The F-4 drone platform was at the request of the USAF. I would think the T-37 would be too slow for the desired mission profile and higher time T-38s could still be "recycled." BAE Flight Systems (my former employer) did design an F-5 and F-106 conversion. Several 106s were droned and destroyed (Including my father in law's old bird on the B-1B chase program). The F-4 presented plenty of airframes that were cost effective to mod.
I spent about 5 years on the QF-4 program at Mojave as a QE.