I appreciate your reflections on the subject.
I'm glad to hear that Classics does value some interdisciplinary work, although I think that you confirm for me that some combinations are more valued than others. Personally, I think I'm left of even the cognitivist. The reduction of Classics education to grammar is frankly numbing. Apologies if that is offensive to anyone. But the historical contraction of Classics to an adjunct of linguistics or to the articulation of grammar - along with the professional bag-checking of everyone's grammatical facility - is dispiriting from a research side. (Yes, grammatical facility is paramount, but that has become the end and not the means). There's only one person outside of a Comparative Lit program that does anything close to what I'm interested in, and I have been told that he is discouraging his dissertation students from following in his interdisciplinary foot-steps.
That said, I may not even do a PhD - or a Masters - at this point.
With Trump president and a Republican controlled Senate and House - and a vulnerable, de-regulated market - I'm not sure if its prudent to do anything else but work while I can.