Of course, the P can play a central role such as in a lot of soul and rnb and reggae etc, when the music isn't busy. Sorry if I wasn't clear.
With the J I have never been able to put it to any good use, whereas I view the P as more a plug and play bass that is either heard or felt. The bridge pickup may as well not be there, that bright nasal sounding tone that cuts through a busy mix is grating to me. When they are both on full, you're right, it disappears. When the neck pickup is on full and the bridge is off, you get something that tries to be a P but doesn't quite convince.