Either it isn't worth it (too much coupling between fab and designs - not the right abstractions), or institutional inertia stops them doing it.
So if they would fab their chips somewhere else, they would be sitting on a huge expensive asset producing nothing. If they couldn't find a productive use for their fabs it would likely mean the end of the company. And if they can't produce their own chips in their own fabs, why would anyone else want to use Intel fabs?
Further, the vertical integration of fab process and chip design is something Intel regards as a competitive advantage. For a long time this was very much true, but it seems the hard work by TSMC and others have made it possible to make top-end chip designs on a merchant foundry process nowadays.
Intel at some point tried to play the merchant foundry game, but it seems it wasn't successful and they shut it down. Which perhaps isn't that unsurprising, considering TSMC, and to a lesser extent Globalfoundries, have been at that game for decades and they're good at it.
So all in all, I don't think fabbing their chips at some third party is a viable approach for Intel. Either they fix their process or they go under. "Go under" not necessarily meaning bankruptcy, it could also mean a massively, hugely downsized company doing chip designs to be fabbed at some third party. I think they're still far away from such a drastic step.
It wouldn't surprise me if the reason Keller left is because he wanted to outsource manufacturing to a competitive third party fab, and got denied.
Keller is an a computer architect, but architecture doesn't matter if you can't physically build the thing.