Basically what we're saying here is any guitar can do Jazz?
I swear I am not trying to be an jerkwad, but the first distorted part of the clip was aweful Jazz tone. I couldn't listen to much of it, I would guess he gets into a lot more of that distortion later in the clip...I don't know what kind of jazz they are doing, but I don't hear any 'great' tone.
I've never understood the "traditional" sound. A muddy neck pickup on a hallow body has never sounded attractive to me. The "Jim Hall sound" with the tone rolled off is even more puzzling. Even more puzzling to me: joe pass, Wes Montgomery, etc, never got a sound they liked with a bridge pickup.
I've been thinking about going back to singles on my Strat for a while now. At the very least I'll go to HSS, perhaps SSS. With all of the guitars I've got these days I've less and less need for three guitars with two humbuckers and more and more need for great Strat tones. The only trick I'm trying to get around is figuring out how I can use a LP/SG style 3 way and still get the most out of all of the pickups. I'm thinking I'd have the 3 way do neck, neck+bridge, and bridge, and then have a switch to add the middle in. Then I'll have another to bypass the 3 way and just do the middle. It sounds like a lot of work, and it is (especially considering I want to put the 3 way in the Alex Lifeson position) but I REALLY dislike the fender style switch.
Tele neck pickups also work for jazz extremely well. With or without the tone control, depending on what you're going for. Telecasters are just great for everything, even some heavy metal.