Is there any specific reason it needs to be "gwylio" and not "edrych ar"?
I don't believe so, but I don't use edrych ar for anything I always use gwylio.
Edrych ar - looking at, to look at
Gwylio - watching, to watch
Right, but I've seen "edrych ar y teledu" or "edrych ar eich hoff rhaglen teledu" many times.
Oh it's correct, I just don't personally use it.