How are you concluding that the P-Bass was the defacto bass used? To start off with pop, three of arguably the biggest influential bands of the 60s - The Beatles, The Rolling Stones, and The Who - were not P-Bass users. Yes, there were several bands that did use them (Beach Boys, Moody Blues) but - more often than not - I see other basses used by influential rock and pop acts from that era: Gibson/Jack Bruce/Cream, Gibson or Ric/ Peter Quaife/ The Kinks, Jazz Bass/ Jimi Hendrix Bass Players, Jazz Bass/ Mel Scahcer/ GFR(he used both), Fender Jazz and Alembics on most of the 70s/80s Dayton-Ohio Funk bands including Aurra, Cameo, Lakeside, etc. In Motown, a lot of the songs attributed to Jamerson were actually done by Bob Babbit who - although he plays both P and J - primarily recorded with a Jazz Bass back in the day. In Prog Rock, its almost safe to say that the Ric 4001 spearheaded that......The "P-Bass Leads The Way In Everything" Mantra is quite a fallicy and has all the credibility of a Facebook Meme (which is also how it gets spread around)
Entwhiste, Wyman, and Quaife all used fenders at some point, P's for both Entwhistle and Quaife. I'll concede your point regarding the J's. Most of the other non-Fender users were one-hit wonders and bnds of the caliber of Tommy James and the Shondells. And what was performed with and what was recorded with often are too entirely different beasts. If we add in the fifties,
which I did and you apparently did not, the balance tips in favor of the Precision simply due to no real competition. Alembic IMO falls into the slappity time frame. From the perspective of now, it's all moot.