"If you could prove any algorithm will halt or not halt, then you can easily make a halt-detection machine that works in all cases."

That is only true if your proofs work from the same one set of axioms. But in reality you can choose different sets of axioms as needed. So it may be possible to prove of each and every algorithm that it will halt or not, but you cannot make a halt-detection machine that works in all cases, because your proofs use different sets of axioms.