I’m not convinced decision based off if he knew he was on fire or not is valid based off the radio calls or not landing immediately.
Aviate Navigate Communicate.
He had made an emergency call and been given priority landing.
Further com’s on fire update would not have changed any response from the tower. They still press the big red button. I guarantee it would have initiated more coms back from the tower when he had his hands full.
less than 1 minute from the initial problem, definitely still trouble shooting and resolving. If I was in that position, updating the tower would have been low priority unless needing the tower to confirm it.