That's one way to define a session player. In 1969 I was asked by an internationally famous artist to sit in on bass on a recording session he was doing. It was my first exposure to the world of "session" players. Not even any charts. We made it all up on the fly.
But I have been asked to record on other projects where I was not a part of a band, but was asked to record the bass part on a project. I was given charts in advance, but no rehearsals. Did that make me a session player? FWIW in all of the recordings I never used a P-bass. I was never asked to use a P-bass by an artist, an engineer or a producer. The only specification that was made was on one session where the artist asked if I would use my double bass and once when the artist wanted a Jaco-style but left the choice of bass to me.
So the original post asked "Why do the session legends all use P basses?" The answer is "They don't, not all of them". But maybe there is a problem with the term "session legends" in that we need to be clear about what we me by "session" and "legends". We are having enough trouble defining what a session player is, and there have been a lot of suggestions about famous freelance bass recording personnel that are known for using basses other than P-type. The very question is bait for disagreement and, judging from the number and nature of the responses, it has been a highly attractive bait.
Interesting - I do get the feeling there's a lot of over generalisation regarding the question of using P basses. I don't even know if it's correct - in fact it likely isn't - I mean some people would have you believe everyone buys Ford cars - well a lot do but VW (and previously Toyota) actually sell more - similarly some people would have you believe tomato soup is the only soup to have but Heinz sells loads of flavours.
It seems to me that some people have a personal preference for the P bass and then extol it as the only Bass - it has a lot going for it but has limitations also.
I'm pretty sure there are lots of other revered recording basses - after all a bass guitar is a fairly simple instrument to record - it's surely isn't rocket science (btw I've always recorded with a Stingray - sometimes with mutes engaged - they give a great recorded sound and sit in the mix just great).