In real life, wouldn’t this be settled by time in rank? At least among nations which consider each other as legitimate peers, as would have been the case for France, Britain, and the US in WWII.
In real life, it would have been settled by force of personality. Soldiers are people, first and foremost, and in extreme circumstances they revert to normal human behavior: some people take charge, and other people follow them.
Also, “time in rank” can mean “number of times passed over for promotion”. It’s hardly an endorsement.
The Air Force has an elite group called Pararescue. They are all enlisted, and they are a member of the helicopter crew.
I assure you that if an officer refused to be rescued and demanded the Pararescue Jumper follow him in combat action unrelated to said officer’s rescue the PJ would render said officer unconscious and drag his body onto the helicopter, with zero consequences for having done so. They will fight, but they are tasked with a very specific mission.
Now, if the helicopter was shot down they’d certainly administer aid, save who they could, and protect the crash site for as long as they could, but again, they wouldn’t follow a shot-down pilot into a frontal attack against a SAM or AAA site.
What if the SAM or AAA site was preventing them from being extracted, and they were the only force available to take it out?
A pilot will typically have zero experience whatsoever in ground combat and will likely have a handgun. A PJ may have a handgun and a rifle. There is almost zero possibility of success.
On the other hand, when a rescue mission is launched there is almost always close air support with it or on alert, and they would have far more success. So it’s best to let them handle it unless there’s no other choice.