Just to clarify - I've been doing "crossfit" for decades - lifting + swimming + yoga, three activities I can pursue into my nineties - and I'm aesthetic AND functional. I just don't do them all at the same time because with old cervical and lumbar herniations (that happened IRL, not at the gym) that'd be a recipe for disaster.
Same with the Crossfit under discussion. Lifting at the gym I can avoid the deadlifts, unsupported barbell rows, lunges, herky-jerky movements and needlessly heavy weights that'd reaggravate old injuries, doing my own thing without being "motivated" or "instructed" to do anything I shouldn't. Regardless of the competence of the Crossfit instructor, I THINK that when it comes to people with preexisting conditions, and even people that KNOW what they're doing, there's a far higher risk of injury with Crossfit. (I'm not going to trot out empirical studies because there's always a contradicting one.) I also think it's irresponsible to rhapsodize over Crossfit while recommending it with nary a mention of the risks. Yes, it's riskier to bicycle or jog on auto exhaust-choked streets thinking you're impervious to other peoples' stupidity. That's why I don't do that, either.
xrichx had the best and most relevant answer, btw. It's primarily the clean & press MOMENTUM that allows you to lift more. Try to replicate that movement in the gym. If you're unable to because the regular gym bores you and you need the conviviality of a group setting and the motivation of others, there's no cure for that so by all means do Crossfit - just be careful out there.