I've definitely given this angle some thought... here's how it shakes out for me.
While I like the idea of a P bass, I don't want to play one.
Glad you mentioned Jamerson. I honestly don't think his big impact had as much to do with his bass as the player himself. By today's standards and what lots of us have gotten used to, to be blunt... his bass sucked. Deader than dead strings on a bass which clearly wasn't set up for a light touch

. Now couple that with the fact that most bass players at the time had no idea who he was let alone what he was playing and it doesn't really slant the ear or the market towards Fender at the time.
Now stack on top of that the tone the Fender was geared towards. Lots of people think that the best specs equals the best product and I've never been on that bandwagon. I've always felt that what sounds good sounds good... doesn't make a difference if it's not flat to 20khz or whatever, what's important is does it make a pleasing sound in the context that it's being used in. There were other choices back then but as much of a Chris Squire fans as I've always been, I didn't personally gravitate towards a Rickenbacker. Gibsons with their mudbuckers? No thanks. Of course that's just me and I know guess who killed on both. Fenders typically don't have the highest highs nor the lowest lows and many of us have found out, much of a surprise to some of us, that that wasn't neccessary.
So I lean towards basses that can carry the load in the mix, not sound wimpy, just carve out that space and demand attention without having to cross the line into being obnoxious. P basses can do that. So can Jazzes. So does my Contrabass.
BTW I think engineers like them because they simply do what they do without requiring a lot of EQ or other sonic gymnastics. The more live gigging I did, the more I understood and appreciated the beauty of a bass that's basically plug and play. Back in the 80's I went out of my way to not gig with my 68 Jazz and some of the "as far as I could get away from Fender" basses were a horror live. I might end up with a sound I could stand by the third set of some shows. Then I got my head out and realized that even if those science experiment basses had been the industry norm all along, I likely would've come to the realization that I liked the Fender tone more. I still feel that way years later. I want good clean clear full range sound but not at the expense of the gonads of a really good ensemble quality (IMO of course) bass.