It wouldn't stand out in the official inquest scenes of "The Menagerie" IMHO - there's no room for "character witnesses" in that procedure, unlike there was in Kirk's earlier "Court Martial" trial.
But if there were former colleagues of Pike aboard Kirk's ship, one would expect them to beam down to pay respects to the poor guy. Not in the teaser along with Kirk and Spock, of course, since one of them didn't know Pike was hurt and the other wasn't telling. But surely later during those events?
Then again, we have no proof Scotty
didn't beam down at some point and find out whether Pike would beep once or twice for his suggestion of hooking a hooch bottle up the IV line.
In that sense, I wouldn't shut out any of Kirk's colleagues from the circle of former friends of Pike, at least not outright. Even Chekov could have met him at some point, what with Pike being involved with training cadets and all.
Except that it happens after Kirk works a control that closes the door. Why would V'Ger wait until the door was shut to materialize the robe?
I'd suspect Kirk would wish to lock the door for reasons of slowing down an invading enemy, rather than for reasons of modesty. And why wouldn't V'Ger wait? Its procedure of sending in the probe was almost agonizingly slow and gradual to begin with. For all we know, the process started with beaming in the artificial endoskeleton, then those micropumps and whatnot to flesh it out, then the skin, and
then our heroes entered to see the final phase where the outer garment arrives.
Why does it "care" enough to create the probe in Ilia's form? The robe would be an arbitrary choice on its part. OTOH, I don't really see Kirk caring.
Decker might rush in to protect the modesty of something that looks like his beloved. Kirk should order phasers set to kill.
wasn't the "Spray Wear" mentioned in the production material for TMP as well as being mentioned by Robert Fletcher in the Roddenberry/Sackett book on the movie's making?
Who are these people, and in which episode did they appear?

(Save of course for Admiral Roddenberry, a famous multicentenarian and holder of several canonically established Starfleet jobs and titles.)
If we're really going to go by canonically unvoiced 1979 intention here, Kirk shouldn't really have a nudity taboo.
Timo Saloniemi