> Probability-zero events happen all the time. The probability of getting any specific value selected uniformly at random from the unit interval (say, 0.232829) is zero.

I would strongly challenge that claim. First, you did not choose that number uniformly at random, you chose it from at best a countably infinite subset, or more realistically, from a finite subset. And secondly, I do not think you can describe a situation where a number is actually chosen uniformly and randomly from the unity interval.