I think the LP tone comes from a combo of all of the things mentioned above. Body thickness, TOM, set neck, scale length, mahogany neck/body with maple cap, etc. The maple cap on a LP is pretty thick to allow for the carve, so I don't think the 1/2" maple top will make it too bright.
While I agree that humbuckers play a part of the LP fat tone, it's not the only difference by any means IMHO. My example; I can get a "fat" tone with my LP through my amp, then plug in a strat with a bridge humbucker into the same amp with the exact same settings, and the tone is much cleaner, brighter, and thinner. I have to redial my amp to try to get the fat back, and it never quite gets as "fat" as the LP, even though the strat has a humbucker.
I think it's an equal part of all of the components in a LP that create the tone. Change any of the individual parts, and the tone changes.