“Saluting military personnel is something presidents are expected to do since they’re at the top of the Pentagon’s chain of command.”
No, they are not. Despite being the Commander-in-Chief, the President is still a civilian and therefore is not expected to salute military personnel. That is the point: The military works for the people, not vice-versa. No president saluted the military until Reagan (and we can make comments about a senile actor thinking he is in the army all day on that one). Since then it is has been expected by the conservative public, but not the military itself. (Some of whom might actually have a problem with someone who never served acting like they did.)
But then I ran into another Brit who thought the First Lady is addressed as “First Lady.” No, she is addressed by her formal last name: “Mrs. Obama.” The term “First Lady” is only used in the third person.
Sometimes it’s gratifying to know we’re not the only ones who can be ignorant of foriegn cultural traditions.