You sound to me like you think you're sarcastically saying something that's just so obviously false that it constitutes a devastating argument against the entire idea of constructive mathematics, but whether the reals "exist" depends a lot on your definition of "exist" and is still an ongoing debate for some definitions of "exist" that are relevant, even without bring constructive mathematics into it. And it's not that hard to draw a perfectly conventional mathematician into an interesting discussion about whether the real numbers deserve to be as fundamental as they are when we can't point to any of them. That is, it isn't just that we can't "construct" them for some definition of construct, but we can't even identify them for any definition of "identify".

Let me highlight again that the question isn't whether they are useful, or a mathematically valid construction; there are all kinds of other mathematically valid constructions far stranger than the "reals". The question is whether they deserve such a fundamental place in mathematics and education. I do recall even in high school that truly understanding the real numbers even at that level was definitely a dividing point for the advanced math class; there are some who really got the quirks, and there are some who never really did and just parroted their way through the tests wondering how the question of "numbers" got so complicated... and it was still a very partial and somewhat ad-hoc introduction to the topic ("Dedekind" doesn't come up, etc.) just enough to try to motivate limits.