$\begingroup$

I suspect that the reason you see more discrete math theorems formalized is just that they're "closer to the ground" than some arguments in other fields, and computer science and discrete math have been historically related, so more effort might be put in in those cases.

I don't think there's reason to believe that there is a correct "proved" result that's established enough to be in a textbook that can't be converted to a computer-checkable proof given enough time, but some proofs/methods of arguing about proofs don't lend themselves to formal arguments.

As you point out, some of the arguments used in algebraic topology or differential geometry might involve arguing about manipulations of certain shapes based on the spatial reasoning and intuition of the reader. There are many intuitive results that would take a lot of time to formalize.

For some of the specifics you mention, someone who is not working in logic/model theory/set theory basically doesn't have to worry about second-order anything, since everything you would want can be done with the first order theory of ZFC. For example, here is a computer-checked proof that a construction of the reals is complete, based on the necessary parts of ZF. And a bit of algebraic topology has been done, like establishing the fundamental group.