That's why for me, constructivism - or in general mathematics with different axioms - are additions to classical mathematics, not replacements. And in that sense, they're fine for me. You'll definitely find people online defending the position that all classical maths is bullshit - I'm not saying that actual researchers behave like that, though.
But I'm also a strict formalist, or more accurately, I think whichever axioms happen to be useful should be used, without much need for there to be a epistemological commitment.
Well yes, it's still a different kind of math. What's not true however is this common notion that a constructivist can only ever view all "classical" math as pure nonsense. In many ways, it ought to be a lot easier for someone committed to constructivist semantics to grok a classical development than the converse - because decision procedures, computations etc. are way more of an afterthought to mainstream mathematicians.