^My take on T'Pau's responsibility for allowing Kirk to put himself in danger is that she gave them every opportunity to leave or back out, but she wasn't going to volunteer any information about what was involved, reinforcing the whole pattern of behavior of Vulcans trying to keep this side of themselves secret.
Her clout as a Federation VIP, which got Kirk off the hook with Starfleet, and the private nature of what was involved, would have protected her from any repercussions for endangering Kirk...and the fact that she had allowed Kirk to get into a situation in which he had to kill or be killed by his best friend, added to the private nature of the proceedings, would have kept her from pushing the matter of Kirk and McCoy having "cheated"...so as far as anybody getting in trouble for this, it was a wash, on to the next episode.
I suspect that some details of the Vulcan mating customs did get out into the public domain after this incident, possibly initiated by T'Pau, to stop something like this from happening again. Hence Droxine's knowledge. And once it was officially out there, it would have been illogical for Spock to deny it when asked.