My employers actually do regularly check employees social media (even had one of HR team mention they noticed I wasn’t active on social media) so it does happen.
The club should not be telling players who they cannot speak too but at the same time maybe a wee bit of education on how to use it and present themselves might be a good idea. Nothing wrong with wishing one of their mates good luck at their new job as bench warmer at a vile bigoted evil empire but maybe have them being educated enough to keep it to a private message so as not too publicly link themselves to said evil empire.