On the flip side, type systems bring at least some sense of proof into programming.
That seems like a weird conclusion to me.
You definitely can't write a "correct" program unless you know what you expect it to do. It's easier to write a specification than a program, because a specification is at a higher level of semantic abstraction.
Is it easier to write a correct specification that correct code? Yes, absolutely, because in a specification I can specify outcomes without having to say how they're achieved, and I can specify results that are invariant over time without having to say how they're maintained, etc.
A specification is essentially a bridge between, on the one hand, a higher-level requirements document that's probably imprecise and/or contradictory and/or incomplete and, on the other, code that has to be completely deterministic.
> [It's easier to write specifications than implementations because] in a specification I can specify outcomes without having to say how they're achieved
It's even easier to write a few "unit tests" that "specify" intended behavior, write code that passes those tests, and then worry what to do in edge-cases when they happen. This way, "correct" isn't a meaningful category, there's only "incorrect in retrospect". This is how most software is made and for most applications I would expect cost-benefit analysis to favor it.
assert(vector.is_empty());
It's even even easier to write a specification that says "if this case, do this; if that case, do that; otherwise, do whatever". Just use your proof language's implication symbol. (this ==> THIS) && (that ==> THAT), done.