I can’t let small things go, his presence in the dispensary in Science blue suggests that he was an orderly or a corpsman. It wouldn’t be implausible for him acting as a guard, too.
In any event, there does seem to be an awful lot of doubling up on duties for a ship with between 203 and 430 people on board.
Dual duties ARE very common in TOS. Many of these were had 3+roles, and at least 2 at the same time.
Spock-Sciences/XO. Sulu-(early on)physicist/(later)helm/botany/hand weapons. Chekov-Navigation/Sciences. Uhura-Communication/(sometimes)Navigator. DeSalle-Navigator/Biology/(later)Engineering. Kelowitz-Tactical/anthropology
I have come to believe that for consistencies sake, the symbols in TOS patches are used precisely to show that a crew person has more than one role. The shirt color shows the most obvious, quickly identifiable set of duties, and the patch shows a secondary set, like in TMP.
For example, early seasons 1 green shirt Uhura has a sciences patch. She is a Command officer, who also has a (technical?) sciences role. In the 2nd pilot, Scotty has a khaki/gold shirt, with another (technical?) sciences patch. Both are red shirts later on.
If Leslie was a medic, acting as security, his wearing a blue shirt but with a support patch makes sense. Adding to that Kelowitz was a blue shirt tactical officer with an anthropology background on the ship.
If Tom Paris was wearing a TOS-pilot uniform, I'd speculate that it would be khaki/gold for support, with a sciences patch since he acts as the ship's field medic.