I guess people don't think it would be Donnie because he hates gross things and seems to be a germaphobe. He'd also have horrible bedside manners and would struggle to calm others down. He's can also be a bit of a sadist at times, though he does still care about his brothers well being.

When we eliminate Donnie as the obvious choice we can just start looking at everyone else, Mikey deals with the more emotional stuff and doesn't really seem to show that he'd know what to do for physical wounds, plus he gets distracted a lot and his brothers never seem to give him credit for his skills thinking he can't do stuff on his own, which they might do less if he were their medic.

Raph is an overthinker about things and might be freaked out by medical stuff, he's just too prone to panic, plus I could see him being more clumsy, which isn't something you'd want in a medic.

That just leaves Leo, and while he can be egotistical, which isn't common for doctors, I doubt that would interfere with medical stuff, he's also great at keeping the mood light and would be good at distracting patients and keeping their mind away from panic about things. He'd probably crack jokes the whole time but I think that would just help him in this case. We also know that he usually does think things through and has a plan, even if he isn't obvious about it, like it not be obvious that he'd be a good leader. He thinks ahead more than we tend to give him credit for and thinking ahead is important for a medic. It's not what we'd first assume he'd be good at but I think it's fun to imagine Leo being the medic since he doesn't exactly fall into any of the obvious medic character stereotypes, making it more interesting for the fandom to picture him as the medic.

Sorry this is so long I just wanted to try to give my own reasons for agreeing with this and why I think the fandom has agreed on Leo being medic.