I'm a bit skeptical of this minimalism (if "induction works" needs to get explicitly stated, I'm afraid all sorts of other things---like "deduction works"---also do).

But while we're at it, I don't think you need to take any mathematical statements on faith. To the extent that a mathematical statement does any useful predictive work, it too can be supported by the evidence. Maybe you could say that we should include it on a technicality (we don't yet know how to do induction on mathematical objects), but if you don't think that you can do induction over mathematical facts, you've got more problems than not believing in large ordinals!