I'm a math prof in the US. Here, I don't find that "physics and last year math lessons do cover it." If they do where you are, I'm envious.

Btw, as much as I know, basic set theory, math logic and proof techniques (e.g. "How to prove it" level roughly) are almost universally delayed till university years. These generic tools are much more important than calculus!

I don't really understand why. This results in many omissions in school-level curriculum, I.e giving formulas with no proofs. This results in kids seeing math as a kind of meaningless and boring symbol manipulation.

Was there ever a discussion around this problem in the US?

The explanation was shallow at best, omitting key definitions, coming up with Newton-style "infinitesmall" hand waiving.

They mostly made sure we were familiar with relevant notation. This somewhat helped in early uni physics-related courses, until calculus caught up.

That was in a math school back in post-soviet lithuania, so russian approach to math schooling implied.