I don't have a problem with it. For example, Pike knows T'Pring, Sarek and Amanda personally; knows Spock has an adopted sister, and also knows about box half brother Sybok. At this point in his life, Spock seems very open about everything. Once he's under the command of Kirk, he's changed, and he is suddenly very closed off.It DOES seem strange that Kirk doesn't recognize the Gorn in Arena now, this hinges on him knowing absolutely nothing about Scotty's service history (whose whole ship was destroyed by the Gorn), him knowing absolutely nothing about the woman who has a massive crush on him (as La'an's whole early life has been influenced by fear of the Gorn) and nothing about the Fed tensions with the Gorn (which occurred when he was literally the youngest first officer in Starfleet on the Farragut) and so on. While none of this is impossible, it's also extremely improbable.
Another thing is that the Gorn doesn't immediately infect Kirk with Gorn eggs to win the fight and be done with it. Maybe the Metrons told the Gorn this was too much of an unfair advantage and forbid this action, but that also is highly improbable in context of the Metrons seemingly allowing both combatants to do everything in their power to win.
James T. Kirk didn't find out about T'Pring until TOS S2 Amok Time and had no idea that either sarek or Amanda were Spock's parents until after they walked out of a shuttle and had fully introduced themselves to Kirk in TOS S2 Journey to Babel.
We have no clue how open Scotty is about his past.
I don't get the idea that because some people are assigned to work together that they automatically know everything in their backgrounds. Not everybody shares everything with their workmates.