I'll throw something else into the convo here: I have played P's mostly for ages live and recording--the "sit in the mix" thing is obviously a known quantity. However...I've been digging on my Jazz lately, and I'm finding that, the both pickups dimed setting is actually the "tightest" most focused sound, and is the most even throughout the range of the instrument and dare I say, the most consistently "sit in the mix" setting. Doesn't sound like a P of course--it's like one giant humbucker spread out across the strings as opposed to one in a narrower configuration--but I wonder if Leo intended that was the "ground zero" setting for the Jazz. Then any fiddling you want to do with pickup volumes is cool, but secondary to the main tone of the bass, and if you wanna go there, deal with the potential hum (like he may have with a Strat...)
Sitting in the mix isn't quite the same as scooping out the mix... Jazz basses actually had less of the fundamental in their tone than P-Basses.
Also, if you're playing a jazz without the individual string mutes engaged, you're actually not playing it the way Fender originally designed it. Eventually he made muting a choice with MusicMan and the adjustable mutes, but when the Jazz Bass was created, it was expected to run with the felt mutes engaged all the time.