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.
Though addition might not be defined on some of the reals, if represent it like that.
But Proof checkers like lean/coq all use Cauchy sequences to represent reals. So they are in effect constructible.
I'm not sure if this answer is relevant, but might as well mention it.
>idea about a circle
As if a circle is a real thing. (Circles do not exist, either - or, rather, they ‘exist’ in a very particular sense of the word, as they are themselves abstractions; round things do.)
Well, yes, exactly. The thing that intrigues me about this discussion is that it seems like mathematicians have a much clearer idea than I do about what the differences between these kinds of abstractions are.