Dancers, gymnasts, Martial artists, and calisthenics guys show up and tend to do pretty well at climbing straight out of the gate. They bothered to learn how to move, and the skills translate at least somewhat.
When it comes to health and longevity, simply going for a brisk walk on a regular basis will realize the lion's share of benefits.
Finger strength, overall strength, flexibility, coordination, balance, and technique all play a massive role as well.
I don't know why the bodybuilders I've climbed with seem to be worse than overweight/obese people I've seen climbing for their first time, but I've had this observation as well.
Who? The heaviest I can honestly think of would be Chris Scharma, Magnus Midtbo, Jimmy Webb and maybe Jan Hojer, but all of them would be considered pretty skinny by normal standards. Scharma and Webb are probably around 6' 160-170 lbs, Jan is around 175 I think but he's 6'3", and Magnus is 5'8" and 158 lbs or so but has incredibly strong fingers.
Climbing is a strength to weight sport, so being heavier will almost always be worse than being lighter no matter what. It's why you don't see as many tall professional climbers as you would expect.
Here is a larger climber who is sponsored by a lot of climbing companies: https://www.instagram.com/drewclimbswalls/?hl=en I'm not sure the extent to which he might be considered a professional, but having multiple sponsorships suggests it's semi-professional for him.
He's not pushing the limits of the sport, but he's promoting a more inclusive attitude. Climbing for a lot of people is about having fun, and very few people will send the hardest climbs, but you can be professional without being elite.
Klem Loskot: 190 lbs
John Gill: 200 lbs
However, None of these guys are performing at a top level today, none of them have sent V16 or higher. Gill is the father of bouldering in America, and was setting standards from the 50s-70s, and claims the hardest problem he has done was about v10, at a time when no one else was anywhere near that good. Bouldering was a lonely pursuit in those days. He was able to hold one-arm levers and do one-pinky pull-ups.
Gill was also plagued by tendon pathologies in his elbows throughout his career, and those issues are more common amongst heavier climbers. Don't let that stop you from trying climbing if you're heavier, amongst mortals it's not such a big deal.
The "not happening" was that my weight distribution feels like it was ripping me off the wall. I just can't get much percentage of my bodyweight supported by my toes because my big back, ass and quads are going in another direction. Turns into an upper body exercise and one-arm pull-ups on a sheer face aren't happening.
Strong people that start climbing tend to use their upper body strength to get up the wall, fat people don't have that option and naturally fall into using technique and their leg strength to get up the wall.
When I started climbing I could do 3 pullups, and now I can do 18 in a set. I don't train pull-ups, just climbing.
Climbing is adaptive, and usually there is more than one way to solve a problem, and one is encouraged to use their body any way they can to get through a section. Many elite climbers have distinctive trademark styles and are instantly tecognizeable just by how they move.
Jan Hoyer is an outlier who posesses both a freakish abundance of power and span. He trained the muscle up to outrageous extremes.
Jan is the only climber to ever throw a 1-9 on the campus board. He performs many outrageous stupid climber tricks throughout this 8 minute video;
Here's Alex Honnold talking about it: https://www.youtube.com/watch?v=kDUeh8nyLT0&t=490s
He demonstrates he can only do 5 pullups (though it's possible he was being modest).
Rock climbing is especially upper body strength intensive. It’s why the guys have ripped upper bodies and tiny legs.
The fat old climber is some inspiration. https://explorersweb.com/fat-old-climber-gordon-robb/
There is a pot-belly 35 year old at my bouldering gym that is better at overhangs than most of the people there, including myself.
I was curious about this so I did a little searching. I found some studies showing that resistance training has lower injury rates than contact sports, and youth resistance training in particular is safer than pretty much any other athletic activity. But I didn't find any studies that covered bodyweight strength training, so I couldn't evaluate GP's claims. Regardless, given the low injury rate for resistance training, I think it's fair to recommend people lift weights, and I mostly agree with your last sentence.
An amateur doing a standard squat, bench press, curl, deadlift etc. routine with manageable weights and good guidance/form is absolutely going to benefit from it. A quick, high intensity weight routine few times a week is probably the best lifestyle change you can make for long term health.