I don't know jack about the history of P basses, tbh.
I own a 1960-vintage Fender Precision bass with a Novak PB-V pickup in it. Also, I own a 50s Classic (reissue) Precision bass. Both are sweet instruments.
After a point (also having an offset-series Mustang PJ bass kitted with EMG Geezer Butler pickup pair), I'm not sure that "best" means much beyond specific goals and preferences. Goals mainly being pickup(s)/tone/output, preferences being first neck profile AND, at least for some, pickup(s)/tone/output.
I'm adaptable where neck profile is concerned. I might prefer my 50s Classic's baseball bat neck to the original 1960 neck by a small margin, as it works well with my wrist position when playing standing up, and I think the Mustang's neck is just a lil bit weird... but I can get used to it.
Ultimately, when it comes to sound, I'll think about pickups first, esp when recording. Then I'll think about neck profile if the sound is a given. Whichever bass that makes me pick up (incl three other non-Fender active basses!), well, THAT's the best bass.
That said, I played on a Fender American Pro II at a brick & mortar store not too long ago when shopping for another bass, and it felt and sounded pretty damn nice. Trying to remember what finish whilst in a glut of basses there, but yeah. For a brand new one, you could most def do worse.