Salutes on special occasions, ok, but not a regular thing - for example, AGT, the scene Picard arrives aboard. I'd have rewritten that scene so that Yar ordered "Hand Salute" as part of his arrival - Trek writers don't understand or often even know about military protocol and pomp, so they leave it out due to lack of knowledge. OTOH, I LIKE that we don't see Crewman Timmy having to salute daily.
Some official honors do seem to be present in TOS: the honor guard in "Journey to Babel," ruffles and flourishes in "The Savage Curtain." X-number of gun salutes are probably so impractical in a space service that they have died out.
As for the hand salute, as Timo
said upthread, the apparent extinction of hats and head-covering in Trek's time may have taken that particular tradition along with it.