Why can't the old man do the CPR, if he knows how to and Cloud don't?
We know it's Because Protagonist. But narratively…
Well, old guy, probably has bad lungs. Running through all possible options:
Red is out from the start. It is possible to perform CPR on a dog. The reverse is not true.
Barret is massive, and has a gun for a hand. So you have issues of exerting exactly enough force without crushing a small child's ribcage, and it is possible to overinflate someone's lungs during CPR, which is not good.
Yuffie…Nothing that says she wouldn't know how, just nothing that says she would. She's really new, so I don't have a solid feel for her beyond the first impressions and forest bandits don't typically have CPR in their repertoire.
Tifa probably knows first aid, but more combat oriented; bruises, broken bones, concussions.
Aerith would be most likely, but I also feel she's be more magic oriented, so she'd have a Materia that can do the same thing as CPR. Drowning is just a status condition, right?
Cloud, as a Soldier, is meant to be a one man army, including medic, so he probably learned in basic training. But he's also got Swiss cheese memory right now and needed a refresher.
That said, this is all me spinning a line. We all know it's because "teehee lip contact".
More seriously, almost anyone can learn CPR (Except Red. Sorry Red) and I encourage everyone to do so.