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There are so many biological processes which are dependent upon ions of lighter metals (upper part of periodic table) such as $ceK+$, $ceNa+$, $ceMg^2+$ and even early transition elements ($ceFe$, $ceMn$, $ceCu$, $ceNi$), but I haven't yet come across dependence of biological phenomena on aluminium. Is this because there is less use of trivalence or is there some other reason?

The metal’s trivalency is certainly not an issue. Iron and cobalt form trivalent compounds (in the $mathrm+III$ oxidation state) in many of their biologically relevant complexes. In fact, in metalloprotein surroundings the actual ‘valency’ (oxidation state) is not so much of a deciding factor; what matters is the number of ligands and the required rigidity of the resulting complex. While some metals prefer octahedral complexes and others tetrahedral ones, many can take either shape and switching between these is inherent to catalytic activity of many metalloproteins. Aluminium has an all-empty d shell and in its compounds is spherically symmetric (only core shells filled) so it should allow for a flexible coordination environment—like zinc.

One thing that aluminium certainly cannot become is a redox catalyst: in compounds it is practically always $ceAl^3+$. But zinc is likewise practically always $ceZn^2+$ and yet used in a number of (non-redox) metalloproteins. So that’s not the issue. In fact, some enzymes’ catalytic activities might (as a thought experiment) be increased when zinc is replaced by aluminium because the latter has a higher charge and thus might be able to better catalyse hydroxide generation. On the other hand, aluminium typically displays slower ligand exchange rates than other comparable metal ions (exemplified by the fact that it is precipitated as a hydroxide when its solutions are alkalised with ammonia) reducing its catalytic utility.

Reading that, one might well be inspired to think of aluminium as a structural metal, assisting in the correct folding of proteins. Again, there is nothing intrinsic to aluminium stopping it from doing this and it should fare comparable to other redox-innocent metals.

Obviously, the enzymes living organisms use nowadays are optimised towards using non-aluminium metals, so replacing one with aluminium will usually result in a less active catalyst or less-stabilised structure. But if aluminium had been used, enzyme structures would have evolved accordingly to allow for the effective use of aluminium in active cores and/or as structural metals.

As already hinted at by the other answer, aluminium has a low bioavailability. In fact, at $mathrmpH,7$ aluminium is predominantly precipitated as $ceAl(OH)3$ but partially redissolving as $ce[Al(OH)4]-$. At slightly lower pH values—typical of the surroundings of most living beings—it is almost entirely precipitated as $ceAl(OH)3$ so practically biounavailable. Of course, specialised ligands can bring it into solution but these need to be evolved first. The problem of silicates further complicates bioavailability.

In conclusiong, I will agree with the already posted answer that bioavailability, especially low solubility at typical pH values are the most likely answer; when compared to e.g. iron, the latter’s redox abilities make it much more attractive for use in enzymes.

One argument put forward has been that aluminum is very poorly bioavailable, moreso than many other elements. Aluminum oxide is very insoluble in water. In addition, any dissolved aluminum that does form in seawater is likely to be precipitated by silicic acid, forming hydroxyaluminosilicates.

From Chris Exeter's 2009 article in Trends in Biochemical Sciences:

But how has the by far most abundant metal in the Earth's crust remained hidden from biochemical evolution? There are powerful arguments, many of which influenced Darwin's own thinking [15], which identify natural selection as acting upon geochemistry as it acts upon biochemistry. I have argued previously that the lithospheric cycling of aluminium, from the rain-fuelled dissolution of mountains through to the subduction of sedimentary aluminium and its re-emergence in mountain building, depends upon the ‘natural selection’ of increasingly insoluble mineral phases of the metal [7]. The success of this abiotic cycle is reflected in the observation that less than 0.001% of cycled aluminium enters and passes through the biotic cycle. In addition, only an insignificant fraction of the aluminium entering the biotic cycle, living things, is biologically reactive. However, my own understanding of such an explanation of how life on Earth evolved in the absence of biologically available aluminium was arrived at by a somewhat serendipitous route! In studying the acute toxicity of aluminium in Atlantic salmon I discovered that the aqueous form of silicon, silicic acid, protected against the toxicity of aluminium [16]. Subsequent work showed that protection was afforded through the formation of hydroxyaluminosilicates (HAS) [17] which, intriguingly, are one of the sparingly soluble secondary mineral phases of the abiotic cycling of aluminium! The discovery that silicic acid was a geochemical control of the biological availability of aluminium, though now seemingly obvious in hindsight, was a seminal moment in my understanding of the bioinorganic chemistry of aluminium, and although it helped me to understand the non-selection of aluminium in biochemical evolution, it also provided me with a missing link in the wider understanding of the biological essentiality of silicon.

Dr. Exeter is one of the few scholars who appears to have written in depth about this issue. Thus, perhaps it is fair to say that (a) your question doesn't have a definitive answer, but (b) the poorly accessible nature of aluminum over geological time due to its interaction with and precipitation by silicic acid is the leading hypothesis.

It's worth noting that when aluminum is artificially introduced into metalloenzymes in place of naturally occuring metals, the resulting alumino-enzymes can retain activity, as a 1999 article in JACS by Merkx & Averill shows.