It occurs to me that we can't really say, since we only have access to the time of the program, which may or may not reflect the actual computational resources expended.

That's a valid point, but it does presuppose exotic new physics to make that substrate, in which "our" time passes arbitrarily slowly compared to the really real time, so that it can solve NP-hard problems between our clock ticks. We would, in effect be in a simulation. Evidence of NP-hard problems actually being solved in P could be taken as evidence that we are in one.