His point is technically incorrect. He can't exclude all the contradicting observations then conclude that all the observations agree. That is a tautology by construction. And it is also unscientific, the physical constants don't have hard boundaries. Each significant digit gets fuzzier the until we just can't guess what the next one is. There are going to be measurements that disagree about the least significant digit.

We can't even be that confident they are constants. We've only had good measurements of them for a few centuries. He's right that proving that would be a big deal, but it isn't true to say that there are no inconsistent measurements.