A Phd would help you a little bit to get selected, in that it's another box to check and shows dedication, academic prowess, and some intelligence (depending on the field of study).

But they are looking for certain things in pilot candidates, and having a Phd will not waive those requirements (some fitness/athletic ability, multi-tasking, teamwork, etc). So you would want to work on the physical fitness...military pilots do not have to be extreme athletes but you don't want to struggle in early training or have that affect your class ranking.

Longer term, the Phd will help with promotions, more so as a senior officer and will open up career opportunities later on if you stay in (War College instructor, acquisitions, R&D).

Be aware that pilots slots go first to academy grads and then to ROTC. So as an off-the-street applicant it may be more competitive to pick up a pilot slot.

Also look at the Air National Guard and USAF Reserves...as a part-timer you could employ your Phd in the civilian world and also fly.