The second distance is 5 diameters not 5 rotations.

So your example starts with a wheel who's diameter is a divisor of pi, which I think it a flawed example (since pi gets divided out in C = [pi] * d)

I wouldn't expect a craftsman to deliver a wheel who's dimensions are specified by its circumference, I'd expect them to make a wheel based on diameter (or maybe radius). We can see that even in today's measurements (e.g. a 8mm bolt has a diameter of 8mm).

So with that, a 1m wheel would have a circumference of 3.14 ish meters (which would then emerge when doing math on the side length of a structure).

My thoughts then go to the units (at top level commenter noted. If they used an arcane ancient unit of measure like the Roman pes (using cause I'm most familiar with that one) to measure their wheel and say the wheel is 5 pes in diameter, their wall will be 5 * [pi] * [rotation count] pes long.

But a pes is .296 meters. So assuming the wall is 10 rotation counts long; we come along and measure the wall as ~4.67 meters long....I don't see pi in that number.

.....so how did the "ancient astronaut theorists" conclude pi was involved in this at all? I suppose that means they knew the units the Egyptians used to build the pyramids? But that doesn't seem very likely.

If you find both circumference and diameter based measurements then searching for a ratio can uncover the unit size of the wheel. Which is probably something you do when you already expect pi to appear.