Kirk's orders supersede his preference, which was to stay away from the Eminian system. The Federation Council ordered Starfleet to go there and Starfleet Command assigns Kirk the job. Kirk might---well, likely---have handled things differently except that Fox (representing the Federation Council) throws his weight around.
Kirk is basically being passive, until Anan 7 declares his crew is to be killed and his ship destroyed.
Whoa! You want to be left to your little game war, fine, but I and those I represent didn't sign any of those agreements. At this point if Kirk and the Enterprise
had still been left to leave then it all would have been fine. But by taking Kirk and his party captive and condemning them to death as well as demanding the Enterprise
and crew surrender to the same fate a line was crossed. That action was an overtly hostile act to the Federation and extremely arrogant for the Eminians to expect Federation citizens would docilely accept it. Can you imagine if they had tried that with the Klingons or Romulans? They'd have laid waste to the planet without giving the Eminians any ultimatum or time to think about it.
The Eminians prided themselves on finding a way to deal with their inherent aggression, but they misjudged thinking humanity would just accept the Eminian's logic.
Make no mistake: the Eminians fully intended to kill Kirk and his crew. No where is it written Kirk has to meekly accept that.
I also wonder what happened to the Valiant's
crew fifty years earlier. It's possible the Eminians got in some likely shots that disabled or destroyed the ship (like they tried with the Enterprise
and then any of the Valiants's
crew on the planet were eventually put to death.