I didn't know Les Pauls could sound bright, actually. At least after reading Filippo's thread sa kabila, naenlighten ako dun. I always thought Les Pauls sounded thick and chunky. Lots of the bands I like used them but I guess I'm just partial to the feel (never played a "good" LP before so yeah, biased) of Fender Jaguars and Jazzmasters (tried a '62 and a number of mexican/Japanese Jazzies/Jags). Soundwise, they just don't do it for me. Functionally, I'm a sucker for that Jazzmaster trem. Suits my needs. Looks-wise, I'd pick offset every time.
Again, I won't say anything about tone considering every guitar has good and bad apples. The older LPs (from what I hear in those compressed youtube vids and read about in articles) seem brighter sounding and a lot more "alive." If that's the case, I might like how they sound from time to time but if you give me free rein to take one guitar from a vintage guitar shop, I'd go for a '59 Jazzmaster or a '62 Jaguar.
Everyone appears to like Les Pauls. I don't hate them but I'm not too crazy about them. Not a fan, really.