I agree with you, except your claim that it proves too much. You will indeed never pick pi or any other specific number you can name. I don't know if you were trying to say my argument leads to those results you consider absurd, but they are in fact the correct results.
One way of interpreting this result is to say that randomly selecting a real number isn't physically meaningful and this does not fit into our human brains well. You can do math on the "random number", but you can't ever actually have one in hand (as a sibling comment to yours points out correctly, it is guaranteed to be uncomputable as well), not even mathematically, so you should expect counterintuitive results. Which is generally true of the real numbers when you get close enough to them anyhow. They are popular for a reason but they have rather more spiky bits and strange behaviors than one might expect from a number called "real".
I think both of you may be thinking of "random number" as something like what you get out of a RNG or something, but random reals are much stranger than random ints from a finite range.
No, I mean that if you start with "you won't land on any given rational" and sum that up to mean "you land on any rationals", that argument also seems to prove that you can't ever land on any irrationals either, so as an intuitive argument it doesn't really point toward why you'll always land on an irrational and not a rational.
("You'll never land on Pi because the odds of picking out the infinitely many digits is zero"- repeat that for every irrational, and naively sum that up, and it looks like you'll never land on an irrational either, which is not true.)
Suppose you construct a fat Cantor set of measure 0.5. You then select a random real from [0,1]. Similar to the rationals, any particular number in the fat Cantor set has zero chance of being selected. Does that mean that the chance of selecting a member of that set is 0?
In fact, the probability you pick an uncomputable number is 1, too.