You can't deny that an anthem celebrating individual achievement matches way better with the American ethos than a song that can easily be sung by anyone. Not that I'm a fan of that ethos, but yeah the anthem is not out of character at all.

The anthem isn't really treated like a "rally", people stand with their hand over their heart and watch a skilled singer sing it.

It's pretty much only sung at the beginning of professional sports games. Maybe government ceremonies, idk.