Kirk and most of the regulars on TOS were pretty brave when death was in the offing. Sometimes a little too brave, from a realism standpoint.
In "The Empath," Kirk, Spock, and McCoy literally fight over who gets to die for the others, and there's never a hint of hesitancy about it.
In "Court-Martial," Finney says to Kirk, "I wouldn't kill you, Captain; your own death would mean too little to you!" Really?
Real servicemen die for their buddies, but it happens in the heat of battle. The classic case is the guy who throws himself on a grenade, and that really does happen sometimes. But on STAR TREK, these guys contemplatively plan to die in 20 minutes without batting an eye.
I think in real life, the most likely explanation for an attitude like Kirk's (practically saying "I must be the one to die, and let's get on with it!") is that he's devoutly religious. But that's not the aired explanation for it, and Roddenberry would never go along with it.
In "Mirror, Mirror," when one guy has to stay behind and probably be tortured in the Booth for it, Scotty says "I'll stay, Captain." Shatner should have said "Great, thanks!" and run up into the Transporter chamber. What an outtake that would have made. And it would reveal that such offers are not always 100 percent sincere. We expect to be refused.