Point 1 is only true for a proper subset of mathematics. This subset tends to be the only mathematics that are taught in high school (or even in undergrad unless you're a math major), which I think is a huge part of the problem. I can't recall the last time I did a proof of something in abstract algebra, category theory, or algebraic topology that actually involved a calculation of any kind, so clearly a facility with basic calculations is unnecessary for those proofs. Instead what is required is a facility with understanding rigorous definitions and abstractions, which is extremely valuable and important, and of which the average high school mathematics education provides essentially none.