>In order to make these arguments less religious, I think it would be worthwhile to adopt Turing's mathematical philosophy, which called for adopting mathematical foundations on an ad-hoc basis (or even no foundation at all). In other words, choose whatever foundation (if any) for the task at hand. This would make it easier to argue that, say, type theory is a more convenient core for proof checkers.
I think you're misrepresenting the category theorists, they're usually the ones arguing for choosing whatever foundation is convenient for a given domain. A lot of the time, this is a type theoretic foundation because a lot of modern mathematics is about pretending you have function types when you don't actually have function types in your category (such as differential geometry) or that all you care about is a "core" set of operations and the rest of the framework you're working in simply gets in the way (such as Hilbert spaces vs. compact closed dagger categories in categorical quantum mechanics). And if you know already knew dependent type theory then synthetic homotopy theory is easier than classical homotopy theory, some proofs in homotopy theory can take several lectures to present and even then it will still be pretty hard to see why they hold.
It's just weird to see people thing the category theorists are the one being impractical compared to the set theorists like Friedman. I can't think of a categorical logician who doesn't have an active line of research outside of logic; usually in topology, computability, quantum mechanics, or algebraic geometry. They're not just logicians but active researchers in computer science, physics and mathematics, logic is a powerful tool and should be applied in these fields.