> An uncomputable real number can be produced by outputting an infinite string letter by letter with each letter getting picked at random.

This certainly outputs an uncomputable real number, given that you have true random input and not a PRNG.

But random numbers from a seed and an algorithm are by definition computable.

And if you can't compute the number, you have no basis for the claim that it's the right number.

Also, good luck computing with arbitrarily long, computed-on-demand digit strings. Consider: how do you know how many digits you need from the input in order to be sure of N correct digits in the output? This might be possible in general, but it certainly doesn't sound like my idea of fun.