First Olympics was 1876.  First playing of national anthems was at the Paris games in 1924.  That's 48 years of no national anthems.  So no, the Olympics have not "always done it."  

While you are correct that the "tradition" didn't originate in the US or with the NFL, it's still a recent development that has waxed and waned through the years in conjunction with a more general sense of nationalism.  The NFL had been around for decades before the anthem became a routine part of the pregame, and it didn't turn into a massive display until the DoD started paying the league for it.