i think im one of the few here that LOVES sg's, i had two in the past my mistake to sold them yeah.., they're amazing guitars, not that heavy but punchy.
You are not alone mate
SG standard is not that neck heavy, I've got both an Epiphone (very neck heavy) and a Gibson, the Gibson Standard is much heavier and is well balanced, almost as much as my strat. It stays in tune no matter what, even when changing tuning from standard to half step down, with 9s and 10s gauge strings... it just doesn't get out of tune, ever! I'm using it with a hybrid 9-46 set of strings lately, and it works pretty well.
SGs are like night and day compared to strats, check them.
As for clean sounds, they've got classic humbuckers, same ones used in LP Supreme models, so no problem in that department. Same bridge, tuners & pickups. Made of mahogany as LPs, just thinner and with no maple top. Much lighter and best fretboard acces ever with the cutaway. Just because they are used for rock mainly doesn't mean they can't play jazz, metal or whatever you throw at them!
If you check an SG check the 61 reissue though. The only thing I don't like so much about mine is that Standards have a 50s thick neck, which is thicker than standard strats & much more thicker than Ibanez wizard necks.