Not canon, of course, but Spock's World
by Diane Duane even briefly discusses Sarek having dinner with the POTUS - a mostly figurehead position by that time.
In the 23rd century, the Federation is ruled by a governmental body that grew out of the principles established by the U.K, U.S., and various other democracies.
By the 24th century, the Federation is still apparently
ruled that way, but in reality, it is ruled almost entirely by disguised agents from all of the Federation's enemies. Almost none of them are aware of each other, and somehow their individual, conflicting sabotages actually result in decent governance!