> For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
Yeah, we teach it. It ends up showing up again in measure theory, assuming that anyone still bothers to teach the mostly useless Lebesgue integral instead of the gauge integral. Measure theory shows up again in probability theory if you're not using Kolmogorov for some sadistic reason and you have to deal with countability.
Otherwise it's pretty much a dead end unless you're in the weeds. You just mutter "almost everywhere" as a caveat once in a while and move on with your life. Nobody really cares about the immensely large group of numbers that by definition we cannot calculate or define or name except to kowtow to what is in retrospect a pretty bad theoretical underpinning for formal analysis.