Many sports teams are filled with non-citizens. That's one of the things that make this country unique.
Patriotism doesn't need advertising!
Having said that, keeping the flag and anthem in international competition is OK.
Please don't try to flame; it's a waste of bandwidth with me, and I won't respond. Just an observation.
From an anthropological point of view: 1) for millions of years human evolution has been driven by tribal warfare, and 2) team sports like football are really practice tribal warfare. So yes, patriotic rituals before battle is a good thing to practice.
All things being equal, if one team (sports or combat) is patriotic as well as religious and the other side is not, the patriotic religious side will win about 2/3rds of the time.
The "filled with non-citizens" claim is most insightful.