That clip in the OP is Errol 'Bagga' Walker, who plays Ps, Js, Steinies, and Hofners, probably others as well. That's his '78 Antigua P that he often raves about, his main studio bass, I think.
For every great and well known reggae bassist that played a J there was one who played a P, so obviously both work great in these settings.
I prefer the finesse of a J in general, it works great in a passive setup with mostly neck PUP and a bit of bridge mixed in, but I'd play a P just as easily. Flats all the way but that's just me- I like the 'old fashioned' sound of a passive bass without much preamp colour or 'active' sounding basses with a lot of boost or less than transparent preamps, and generally I like the vocal quality flats can have played a certain way. I also prefer a bit of dirt to the super clean modern tones a lot of contemporary bassists get these days. Just preference, call me a stick in the mud.
I've never got satisfactory tone rolling off the tone button on a passive instrument. I prefer it slightly rolled off and to roll off the treble a lot at the amp, though not so much you lose definition. It also accentuates the mids a bit too much on a P and reduces a J to mud, IMO. Because the bass is usually so high in the mix the lows need very little boosting, you just lose punch and end up with inarticulate mud. The real power usually lies in the upper bass/low mids anyway. It's a balancing act between thump and articulation, punch and lows and when you get it right it's glorious.