If you do not defeat Flayn with Byleth, Flayn is actually killed. Defeating her and Seteth with Byleth allows them to live.
When I played that chapter I had no idea that was even possible. I initially killed Flayn but decided I wasn't happy with that result and Divine Pulsed it away. I then ran Byleth over from the other side of the map to try and talk with her. That doesn't work. I tried to see if their in battle conversation would change anything, it didn't seem to. I couldn't think of anything else and eventually KO'd her with Byleth to end the map. I was shocked that she didn't die because I wasn't expecting anything special to happen.
She didn't give me back my March Ring so my efforts were ultimately for naught. ---
Link isn't the only one who kicks ass. Congrats to Advokaiser for winning the CBX Guru Contest!