If social media does indeed lead to less exercise as research suggests-- which, as you explicitly agree would make things worse-- it is a boogeyman.
We don't outgrow that. Adults are very status conscious too.
Social media takes every challenge that kids face and amplifies it.
My anecdotal experience is two daughters whose normal teenage challenges have been made worse through social media.
Social media could have been a utopian technology bubbling up unique experiences to cultivate hobbies and interests in the young. Instead the algorithms cater to our base instincts and is amplifying the risk of turning the next generation into mindless addicts.
I don't see why any kind of urbanization or improved communication/infrastructural technology wouldn't have this effect. The more people you can reach, the more people you have to compete with.
Social Media isn't the boogeyman. It's that no matter how hard you try, your life will never be as good as what is usually portrayed through these channels. You are swarmed with people who lead far better lives than you do, have way more fun than you do, and so on and so forth. Your only escapism at home, in a pandemic, is to go on the internet where you're spammed with these successful people (posers or not, doesn't matter) selling you things by showing off what they have.
So no, it's not social media that damages teenagers mental health. It's worse than that. Ignorance is bliss? There's an argument to be made for that.
It starts before they even hit teens. That YouTube channel of the kid unpacking toys and other things is the kind of early stage precursor to things to come. The kids watching this viscerally live through him for some years, until it dawns on them that hey, wait a minute, he has all those toys and I don't have anything.
It's no wonder exercise makes things better - it's a great distraction from the illnesses of the world. Assuming that children are somehow not aware of it, or are not susceptible to it, is being naive at best.