That would have been a distinct improvement, yes. Something that the TOS films didn't do. Kirk's final answer to the Kobayashi Maru, when you can no longer cheat death. Oh, if only.
It's arguably what Generations
did at the beginning....
Yes, except that Kirk didn't have any clue he was about to die. His intent was to rig the deflector and get the ship out of there intact.
My thought is that it would have been very dramatic, and a very poignant moment, for Kirk to realize that he was in a no-win scenario, the very scenario he had always denied could exist, and make a final, deliberate sacrifice of himself so that others might live. In a sense, he would beat the no-win scenario, but only by sacrificing himself.
And, yes, one could argue that story arc had already been done in TWOK with Spock. But (1) Spock never had a problem accepting that there are no-win scenarios like Kirk, and (2) Spock was
resurrected, meaning it really wasn't a no-win scenario.