How about generating an infinite sequence of bits to generate any positive integer (in binary) with uniform probability? Sure it will take infinite time, but so may the method in your note. As another comment here points out, it would take infinite time to write down a purely random number anyways.
The method I described gives nonzero probability to every positive integer. The method you describe a) will never terminate and thus b) gives zero probability to every positive integer (since there's no stopping condition, even if you ever get to a particular integer you're always going to generate another digit and move on past it).
The essential issue is that probabilities have to sum to 1, but you can't just give the same probability to an infinite number of things because there's no number for which p*infinity = 1. So the only way to get an infinite sum to equal 1 is if the probabilities are unequal, and in particular the probabilities have to go to zero in the limit (e.g. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1).
Parents method produces a result in finite time with probability infinitesimally close to 1. Yours produces a result in finite with probability infinitesimally close to 0. That is the greatest measurable difference in all of mathematics.
In other words, on method basically always succeeds, the other never succeeds (not "fails", but "never succeeds")
