For a second I was as sad as everyone here, before the sense of reason hit me. Like sure it was sad, but there was no real reason to let any of what transpired happen. For a genius, how did she think breaking her hands was in any way a better solution than just asking? How did the fact that D.Va would still be put into service not cross her mind, she's not the only doctor capable of high end prosthetics.
Disagreed, I feel that events of the story do make sense.
Tbf, this wasn't just a lover's quarrel, this was going to be a literal war. D.Va strongly answered against the Doctor's ideology (because of her pride) so that the doctor probably believed that D.Va was unmovable in her stance.
Also, D. Va is a pro gamer, her hands are such a giant part of her life. If D.Va lost her hands then she would probably fall into a depression and stay away from the war. Her talents are greatly neutered and she probably wouldn't be drafted into the war, which is what Mercy intended when she broke her hands. The only reason why D.Va ended up fighting anyways was probably to find Mercy again, that's how she was able to cope with the loss of her hands and use obviously inferior mechanical hands.