This has been a fun thread to read.
I get why people always bring up Edelgard. Byleth has close familial ties to the monastery, and she's involved in a conspiracy that leads to the protagonist's father's death. No, she didn't want it that way, but I think in most countries, once you enter into a conspiracy, you share culpability for outcomes.
The death of Jeralt is a "loud" thing that makes an impression on first-time players. There's even different monastery music for the month. Continuing to work with her shitty uncle and TWSID was a decision. Maybe necessary to achieve her objectives, but not necessary. It's an end to a means.
Like I said, these choices are all kind of "loud" and invite discussion (perhaps too much, but I think the designers intended this). I think these events and others make it possible to overlook how flawed Dimitri and Claude are. As the OP states, they cannot escape their pasts or circumstances, not fully. They, like Edelgard, either forgo or bungle diplomacy when tens of thousands (almost certainly more) of lives are at stake. They are all equally flawed and could have spared countless lives if they had been better listeners.
I agree that Three Houses is a tragedy. I get confused by the many posts arguing for which lord is the better person. They all make bad decisions and only succeed because the protagonist chose them.
That isn't to say I don't like the characters, I do. I even have a favorite. But OP did a good job of distilling a lot of my thoughts on the subject.