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It claims there are three options of which none of them are satisfying.

Circular argument doesn't prove anything because it's just when the premise is the same as the conclusion.

x ∵ x

Infinite Regress isn't clear on why we'd need to do this let alone even possible to implement so it also has no proofs.

x ∵ ... (never ending chain)

Foundationalism is different in that it's actually useful. Mathematics is its epitome. Math has tons of proofs. Math isn't only cohesive but also adhesive. What more do we want?

y ∵ x; where x is presumably true

As for it's criticism: Why start at claim x?

Because presumably, it's either true (via self-evidence or empirical research) or it doesn't matter because sometimes we only want to see where the premise takes us.

Maybe an interesting take on my question is: how do we know it's a trilemma?

Circular argument

We know it's a trilemma because the argument is founded on logic and proofs, and all proofs will end in either circular logic, infinite regression, or a foundational assumption.

Infinite regress

You can always break a proof into parts. Those parts get simpler and simpler. Keep breaking them up long enough, and all parts will eventually become circular logic, infinite regression, or foundational assumptions.

Foundational assumptions

We know it's a trilemma.

All kidding aside, we find this pattern is generally true. Logical arguments end up in one of these buckets. Sometimes you can change which bucket it ends up in (such as calculus, which permits us to replace some infinite regression arguments like Zeno's paradox with foundational assumptions about how limits behave), but they end up in one of these buckets in the end.

The main value of the trilemma is to provoke thought. Its to make you think about what must come of rational thought. If it does so, then it has done its job. Proving it "right" however, is more difficult. The whole point of the trilemma is that people who believe in it are not satisfied with the result of any attempt at a proof.

To prove it, you would have to select what concept of "proof" you wish to explore and what tools you are willing to consider in the proof thereof. Those are rather personal, which means there's not a one size fits all answer, besides the satirical version I wrote above.

I would argue that you have a clear opinion as to what proofs you find appealing. You appear to find the foundational ones appealing while the others feel like "not an argument." Consider, however, the infamous Turtles argument:

The following anecdote is told of William James. [...] After a lecture on
cosmology and the structure of the solar system, James was accosted by
a little old lady.

"Your theory that the sun is the centre of the solar system, and the
earth is a ball which rotates around it has a very convincing ring to
it, Mr. James, but it's wrong. I've got a better theory," said the
little old lady.

"And what is that, madam?" inquired James politely.

"That we live on a crust of earth which is on the back of a giant
turtle."

Not wishing to demolish this absurd little theory by bringing to bear
the masses of scientific evidence he had at his command, James decided
to gently dissuade his opponent by making her see some of the
inadequacies of her position.

"If your theory is correct, madam," he asked, "what does this turtle
stand on?"

"You're a very clever man, Mr. James, and that's a very good
question," replied the little old lady, "but I have an answer to it.
And it's this: The first turtle stands on the back of a second, far
larger, turtle, who stands directly under him."

"But what does this second turtle stand on?" persisted James
patiently.

To this, the little old lady crowed triumphantly,

"It's no use, Mr. James—it's turtles all the way down."

Now you don't have to agree with the opinions in this story, but you must admit that there something worth calling a "proof" here, and it ends in infinite regress. As a thought experiment, how would you go about responding to this? Would you be inclined to tell the little old lady that it is not valid to use infinite regression in proofs?

Consider how we model numbers with Peano arithmetic. We always have an axiom of induction. We may call that simply a foundational axiom, but if we look into why we think it is a valid axiom, it starts to look an aweful lot like an infinite regression argument. We just tucked it away inside a foundational axiom so that we didn't have to worry about it polluting the rest of the proofs. So, perhaps, in a way, such fundamentals of arithemtic simply prove that infinite regress is a valid method of proof in some carefully constructed circumstances!

You have misunderstood the point of the Munchausen Trilemma. It plays a key role in the process of philosophy showing that none of our beliefs are justified knowledge, per the standards of "reasoning".

Most people hold that they have knowledge and beliefs based on justified reasons, and that beliefs SHOULD be justified, and knowledge isn't knowledge unless it is justified somehow. The term for this is the Principle of Sufficient Reason: https://plato.stanford.edu/entries/sufficient-reason/

In its first usage, by Spinoza and Leibniz, the PSR basically called for a proof, before one could satisfy it.

Another key term in the history of knowledge is Justified True Belief -- the criteria that knowledge is only knowledge if one has both justifications, AND it is true. https://plato.stanford.edu/entries/knowledge-analysis/

However, most empiricists since Locke have been indirect realists -- holding that the world is only inferred, not known directly. And if we cannot know the world directly, then one cannot EVER know of a belief about the world is TRUE or not, and JTB is unachievable. https://www.iep.utm.edu/perc-obj/#H2

In practice today, most "reasoning" people follow a softened version of the PSR -- where "sufficient" reasons are just "supporting justifications are stronger than refutations".

However, one can apply the Munchausen trilemma to all "supporting justifications" to challenge what THEIR support is. And the answer, in every case, because we cannot complete an infinite series, will lead to and UNJUSTIFIED supposition, or a circular argument.

If beliefs need to be justified to be reasonably held, then all those justifications, to be held reasonably, must themselves be justified. But they are not and cannot be. The Munchausen Trilemma refutes all claims to be "reasonable" or hold "justified beliefs" based on our current standards of knowledge or reasoning.

The response among many philosophers has been to embrace larger networks of supporting assumptions, this is the coherence reply to this dilemma. https://www.iep.utm.edu/coherent/ The reasoning is that while a simple circle might be a fallacy, a complex web of justifications is not. The name of the Munchausen Trilemma, which critiques the circularity of coherentism, is a ridicule of this claim. Baron von Munchausen cannot pull himself out of the mud, or pull his horse out of the mud, but by pulling on his hair, then lifting himself this way, he was able indirectly to lift his horse through the stirrups! Make the circle big and complex enough, and one can do the apparently impossible with it.

If no beliefs are justified by our current form of rationality/reasoning, then philosophy and knowledge must both become non-rationalist. The response to this has varied between Europe and the US. In Europe, it has been an embrace of radical relativity -- IE postmodernism. In the US it has typically lead to pragmatism, where one can justify beliefs based on the pragmatic criteria of their utility/effectiveness.

Elaborating on the consequences of the pragmatic re-evaluation of reasoning and rationalism. This involves accepting that truth is not an absolute -- but only "approximately true", and that reasoning is not itself justified, or always valid, hence logic problems such as circularity, or unjustified assumptions, are not fatal problems for a assertion or justification.

Hence flawed justifications, such as the Thomist "5 Proofs" of God -- showing they are not actually "Proofs" does not refute Thomism as a POV. One needs to evaluate its pragmatic effectiveness/value in explaining the world, and the revival of Thomism in recent years shows it is pragmatically useful.

And while Descartes' Foundationalism is actually questionable (there have been challenges on selfhood, much more widely accepted challenges on God, and in particular challenges to the trustworthiness of our reasoning) he still assembled a pretty good foundational case, and much modern thinking is still indebted to him due to the usefulness of his argument.

The cohenrentist response to Munchausen's Trilemma may be logically circular, but the argument that we HAVE to be circular to build a worldview at all, is a highly useful and powerful one anyway.

Similarly, the entire basis of the validity of the scientific/empirical method to attain knowledge is an effectiveness argument -- IE it uses empirical justifications to validate empiricism -- in fairly explicit circularity. Yet science is demonstrably useful, hence the logically invalid justification is -- pragmatically still valid.

The trilemma is about justification of a given proposition. Any justification, so the story goes, takes ultimately one of these forms if faced with skepticism.

Therefore, the third option is about people who answer to the question "But how do you know that x really is true" dogmatically, e.g. with "Because it is", "Because I say so", etc.

Ultimately, the answers "It is self-evident" and "Empirical research shows it" are, if taken as an end-point of justification, arbitrary since the skeptical objections "Why do you assume self-evidence here, I do not follow" and "What are the standards of this research, why should this be a particularly good standard for the truth of the proposition" are still valid.

Sure, we accept that for statements about the world, empirical research is probably the gold standard and brings us as close to the truth as we might get, but that's not the setting in which the trilemma is supposed to work.

It is about forms of justification, and ending the chain of justification at an arbitrary point is dogmatic, no matter how well-justified the point itself might be. If I claim that this justification was unquestionable and ultimately justifies the whole chain as true, I am dogmatic.

The first and main point to understand about the trilemma is that it is an argument. As such, its function is to convince other human beings, at least to the extent that they are rational.

The trilemma is an argument about knowledge and for this reason is often misunderstood as proving the impossibility of any knowledge.

This, however, is a logical impossibility. Either the Trilemma is itself knowledge or it isn't. It couldn't possibly be knowledge, however, since if it was, it would be a counterexample to its own assertion.

And if it isn't knowledge, then, why should we care about it?

We need to care because the Trilemma is effective as a rational argument against a certain idea of science as objective knowledge the physical world.

Any idea of science as knowledge of the real world which assumes that subjectivity can and should be entirely left out of science is indeed condemned to an infinite regress. And a good theoretical model of this is indeed the theory of Justified True Belief.

Justified True Belief either leaves you not knowing whether a justified belief is really true, because of the circularity of the theory, or requires that you produce a second-level justification to justify all the elements coming into the initial justification, and then of course to produce a third-level justification to justify all the elements coming into the second-level justification etc., ad infinitum.

Justified True Belief actually describes quite well the history of science, where each generation of scientists produces a new, more detailed justification without being able to say whether the last science is actual knowledge of the real world.

The Trilemma is thus effective against any Justified True Belief view of science as knowledge of the real world. It is effective, however, only to the extent that it is a convincing argument. It is still not, and won't ever be, knowledge.

As a convincing argument, it probably has some effect, up to a point, on how people think about their own practice, scientists in particular.

However, the fact that it is a convincing argument about the absence of knowledge of the real world, in particular scientific knowledge, doesn't say anything at all as to the possibility of knowledge or indeed about whether we know at least some things or not.

Suppose you know X. What the Trilemma says is that you cannot justify that you know X. However, what the Trilemma doesn't do, emphatically, is to prove that you don't know X. Again, the Trilemma is not knowledge and therefore isn't knowledge that you don't know X. It is not even an argument that you don't know X. It is only the argument that you cannot "satisfactorily" justify that you know X. So, if you know X, of course you do, and the Trilemma be damned.

Still, whether you really know X or not is of no interest to anybody but you. What matters to other people is to know themselves whether you know X. However, the Trilemma is this very convincing argument that whatever you will ever say, other people won't be convinced that you know X. And why should they?