Though it's weirder than that. If you can actually put a number on the number line, then you have a computable number. And those are countable.

Uncountable reals only exist as abstract thought experiments.

You are mixing up definitions. Pi is computable, meaning we can calculate one by one its digits, but is not rational. If you can compute a number in a finite time, you can put it om the number line, but the only numbers that can be finitely computed are the rationals which are countable

You can put pi on a number line just fine, though. You can pick any range of the number line to look at, and determine pi's place to arbitrary precision in finite time.

With "uniformly random real between 0 and 1" you can't do that.

More relevantly to infinities, the program that spits out the digits of pi is finite. We can easily count through all the computable numbers, including every number you can specifically name, albeit in a weird order and with duplicates.

It's not a bijection, but if you have a computable number you can derive a natural number that represents its program, and if you have a natural (or rational) number you can derive a program that computes it.

> Uncountable reals only exist as abstract thought experiments.

Technically, sufficiently large natural numbers, such as 10^10^10 also exist only as thought experiments.

Technically, if we require the number to refer to some number of objects, there is such as thing as a largest natural number - it is the number of all particles that exist in the universe. If you add +1 to it, the result of that addition exists only as a thought experiment.

As natural numbers get huge, they become gradually harder to use. Though no, I wouldn't require a number to represent some physical quantity, as long as the number itself fits in some kind of moderate physical boundary.

But real numbers are wonky right from the start. Picking a single random real out of a range is infeasible.