Like I can say a plate is round, an objective fact, but the conception of roundness, mathematical roundness, is something else. Certainly what they do would not be affected by a philosophical controversy. This is true in many other fields. To justify philosophical activity one must go to a deeper level, for instance as in Socrates' remark about the unexamined life. It's pathetic to be a mathematician all your life and never worry, or think, or care, what that means.
Many people do it.
I compare this to a salmon swimming upstream. He knows how to swim upstream, but he doesn't know what he's doing or why. The philosophy of mathematics is very pertinent to the teaching of mathematics. What's wrong with mathematics teaching is not particular to this country. People are very critical about math teaching in the United States nowadays, as if it was just an American problem.