In 2004, I produced and played on a 3-song demo by a singer/songwriter. At the studio, they had a Precision Bass leaning up against the wall in a corner (no case), ready to be pressed into action at all times. I used my Jazz Bass, but would have switched to the Precision if the engineer took too much time to get a good sound from the Jazz.
Back in 1988 or so, I recorded with a band and used my '83 Elite II Precision Bass. The engineer rented it from me for the following days' sessions, because they were going to have another, younger band back in, and whatever non-Precision the kid was using just wasn't making it.
Then you had the Pixies' producer (Gil Norton?) basically ordering Kim Deal to run right out and buy a Precision because her Aria sounded terrible. I know this because a friend and bandmate of mine sold her the Precision when he was the guitar department manager at E.U. Wurlitzer in Boston.
So, play whatever you want... but there *is* some truth behind the supposed "fairy tale."
They just *work.*
Having said all that, I do understand one part of the argument that only old timers continue to believe that only Precisions work in the studio... Before everyone could record on a laptop, at their leisure, you were at the mercy of very expensive recording studios where every minute counted. If you were paying for studio time, you scraped together every penny (including change under your sofa cushions and car seats) to afford it. They would take a while getting the drum and guitar sounds, but if you had a Precision, it was quick-- like in a minute. And if you were a journeyman session musician, who wasn't paying for the studio time, you certainly didn't want to risk being "that guy" who wasted someone else's money by using a labor-intensive bass.
These days, with home recording and therefore no studio fees, we all have the luxury of using whatever instrument sounds and plays like "us." It's a good thing.