I wouldn't really consider Weyl as an example of a modern day critic, considering he died in 1955. I think you'd be hard pressed to find a modern mathematician that rejects the diagonalization argument, outside of finitists, which are pretty far from the mainstream. I mean the proof that there is no bijection from a set to its power set is pretty straightforward and constructive. I'm pretty sure the proof is valid even in intuitionist logic.