Edelgard wasn’t wrong, but she wasn’t necessarily right either. I agree that the church was corrupt at that time; Rhea was corrupt, and you could argue that Seteth and the knights were as well. But Edelgard went about correcting those wrongdoings in the wrong way. The war was not the right path to take; it may have seemed like the only path to take for her, but it was wrong. With Byleth as the new archbishop, while not everything would be fixed, a lot of the church’s wrongdoings would be solved.
There’s also a problem with crests. I don’t think that starting a war over it was a good idea, as crests only affected the nobility. And, even then, the current generation did not care about the crests in the same way that previous generations did. Therefore, I believe that her second motive, the crests, was also not a reason to start a war.
This is my take on the issue, if anyone else has input I’d love to hear it!