I believe the Lua creators' claim on this is not that it's necessarily a good API, but rather that it's "just" a better one than any other they know of :) (Or at least knew of at the time when they created Lua. But I think if they learnt of a better one in the meantime, there's a high probability they would try it, given that they're not afraid to break backwards compatibility when they see value in a change.)
Another problem I've with the stack API is that it's insecure by default. If you push too many things it will result in a C stack overflow. I'm not sure how much performance this gives, I mean, the inability to grow the stack automatically looks like an optimization matter... But at least, panic when we reach the max stack instead of smashing the C stack.