There's no problem at all with being a mathematician or a philosopher of science. I'm a physicist, and I don't think any of my colleagues would argue that these fields should go away or that physicists shouldn't work in them. Emmy Noether is a great example of how people outside physics can help develop new physics.
But... relativity wasn't accepted until it was tested. Neither should any other theory coming out of advanced mathematics. Simply being around for a long time is not enough to move a set of math from clever speculation into physics. We've been down this path before. Allowing foundational theories to be integrated into the rest of physics without verification might end up fine, or it might waste the careers of a generation of physicists. Today, that also might mean many billions of dollars of funding and significant public trust.
You say this like there's some cabal deciding on 'allowing foundational theories into the rest of physics without verification'.
If you look at the Dec. 2014 Nature article that sparked the NYTimes article, you'd see that the concern there isn't even about the conduct of science itself -- it's about the worry that apparent dissent among scientists will fuel anti-scientism. So we'd better work out these 'what's experimentally verifiable' questions far away from the inquiring public.
There's no real worry that somehow the world's best and brightest physicists have forgotten about falsifiability.