Could an NP-hard problem have a mechanical or physical solution method?
No.

Unless $P=NP$, it's unlikely that any mechanical process can solve an $NP$-hard problem efficiently. Given a mechanical approach to solving a problem, we could run a physics simulation on a computer to get the same result.

Of course this depends on being able to simulate what happens in the real world efficiently. In order for "mechanical polynomial time" to be stronger than "polynomial time" you'd need to find some physical process that can not be simulated efficiently. One possible candidate would be quantum mechanics but that doesn't seem to be what you're asking about (and it is speculated that even quantum computing doesn't allow one to solve $NP$-hard problems efficiently).