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My understanding is that air resistance is where most of the pedal power goes to at high speeds, and that this is a concern not least for the wheels: What you pay for in a road bike wheel is aerodynamic and/or light wheels, not so much to other losses like rolling resistance and bearings.

So if air resistance is the thing to optimize a wheel for, the wheel should be tiny. Obviously in width, but also in radius, because even shortening the length of a surface reduces boundary layer drag, and the top half of each wheel is basically a surface that moves faster than the rest of the bike. Has anybody thought of this?

Postulate: The faster angular speed of a small wheel is irrelevant to air resistance: If the forward speed is given, every point on the wheel at a given fractional wheel radius has a speed compared to the ground that is independent of the wheel radius. In other words, shrinking the wheel does not speed up any point on its surface; you just get less surface. Which should be good!

Some speed record bikes have had small wheels, including the current record holder.

The boundary layer drag (skin friction) is pretty small at large Reynolds numbers and even if still significant, most of it does not happen on the wheels, but also on the frame and on the rider, so the wheel causes only a small part of the drag (and even of the boundary layer drag). For bluff bodies, and a bicycle (and a rider) is a bluff body, the decisive part of the drag is the form drag, not the skin friction.

As explained in other answers, all other practical factors favour the design of bicycles as we know them. Especially for off road bicycles the aerodynamics is not very important and you can see how bike are designed for handling other capabilities.

Road aerobikes don't really choose to use smaller wheels (although 650 mm wheels are readily available). What is MUCH more important to the drag of the wheels is the tyre width, because that directly changes the form drag. The wider the tyres, the larger the form drag. Other forms of drag are obviously different. I DO know that racers use larger tyres for today's races, that does not invalidate the argument (I use 40 mm tyres myself for touring and gravel). But thin tyres DO offer lower aerodynamical (not rolling) resistance.

Assuming a bicycle with a conventional rider position, the rider cannot be positioned any lower because there has to be a certain amount of clearance between the cranks and the road. As wheels shrink, the frame has to extend downwards to reach the axles, so you are not really removing structure that causes drag, you are replacing it with something else.

Smaller wheels have higher rolling resistance, at some point that become significant compared to aero drag.

There are a host of other practical reasons, for instance:

As the driving wheel decreases in size the gear ratios have to increase to maintain the same distance travelled for a turn of the cranks. This is why folding bikes have huge chainrings.

Small wheels are less stable, as anyone who has ridden a Razor scooter knows.

You also have to count for stuff you can get over - that's why 29-inch wheels are so popular with the mountain bike crowd these days. They suck at cornering, and that's why they came up with 27.5's. Smaller wheel diameters are great for accelerating quicker, hence the popularity of 650C wheel sets for triathlons. Scooters have tiny wheels because they ideally are ridden on paved roads or sidewalks, and can therefore get by without hitting any major potholes or bumps. At some point, people got together and determined an ideal threshold for wheel size that would prevent accidents by their sole ability to get over stuff. That's the best explanation I've got!

20 inch wheels accelerate faster than larger wheels like 700c. However, the distance traveled is less at the same gear ratios.

Have you ever played with a toy Gyroscope? The same principal is in play with a bicycle. A small wheel would have a similar affect to not moving, i.e. not stable.

As a larger wheel spins up, it keeps the bicycle upright.