You're just covering my third paragraph. Yes, everybody is a philosopher because we don't have the tools to do away with philosophical arguments entirely yet.
Once a mathematical proof has been verified by computer, there is no arguing that it is wrong. The definitions and axioms directly lead to the proved result. There is no such thing as verifying a philosophical argument, so we develop tools to lift philosophical arguments into more rigorous systems. As I've shown earlier, and as another commenter added to with incompleteness, this is a common pattern in the history of philosophy.
It is not necessary to solve the halting problem to show that a particular lean proof is correct.