This is definitely an out-of-universe answer, but in an interview with Oscar Isaac (who plays Poe Dameron) in GQ magazine titled, “Star Wars Actor Reveals Which Character Was Supposed to Die in The Force Awakens” the actor sheds some light on some of the awkwardness of this moment—and it’s “magical” resolution—in the final film; bold emphasis is mine:
…Abrams had pitched him the character of Poe Dameron, a badass fighter
pilot battling against the remnants of the Empire.
“He’s amazing!” said Abrams.
“Sounds good!” thought Isaac, whose first experience in a movie
theater had been seeing The Empire Strikes Back.
“He opens the whole movie!” said Abrams.
“Sounds great!” thought Isaac.
“And then,” Abrams went on. “He dies.”
“Oh,” thought Isaac.
“I’d done that before,” he told me later. “Set up the plot for the
main guy and then die spectacularly.” (He had played just such a role
in The Bourne Legacy.)
Then he goes on; again bold emphasis is mine:
“I went back home [to New York], and I thought about it,” he says.
“Then I wrote him and said, ‘Okay. I’ll do it!’ I figured it would be
a cameo: I’ll come in, do my thing, and maybe it’s actually better not
to have to sign myself up for three movies.” By that time, though,
things had changed and Abrams soon wrote back: “Never mind. I’ve
figured it out. You’re in the whole movie now.”
So at the end of the day, it seems that Finn would have helped Poe escape, Poe would then pilot the TIE fighter with him and Finn to Jakku, and then the whole question of “What happened to Poe?” would simply be… He died and Finn is the one whose story goes on.