No, Gene had a concept that "everyone was an officer in TOS" but this never made it on screen and was therefore never specifically canon (though some of the more recent series, particularly LD and to a lesser extent DSC and SNW imply that this might be the case.
There were prominent enlisted characters in every one of the "classic" series (Rand, Burke and Samno, O'Brien, Tarses, most of the Starfleet engineers on Deep Space Nine, most of the Maquis, Culter, Daniels and all the MACOs save the CO for instance).
Either Rand went to the Academy later in life, after Kirk's Enterprise, or Sulu gave her a field promotion to Commander to hold the position of XO on the Excelsior. If she chose to leave the Excelsior, then the field promotion would have gone away.