Considering the Kingdom didn't know where Rhea was after the time skip and Dimitri was leading the Kingom to war out of bloodlust/revenge and not as a holy crusade, I don't think the church was a main motivator for them after that battle.

The Kingdom and Alliance were obligated to side with the church because another country attacked a neutral military academy while their future rulers were present. Edelgard knew it was an act of war against more than just the church, and she knew she was forcing the entire continent into war, not just the church. You're probably right that the kingdom would've sided with the church anyway, but Edelgard created a situation that forced them to side with the church.

Edelgard could've become friends with Dimitri and Claude while at Garreg Mach, explained to them why she thinks the church and the crest system are detrimental to society, and then worked together with them to dismantle it more peacefully, at the very least ensuring they don't side with the church, but instead she goes full fascist and decides to conquer the continent because she thinks she knows best and she refuses to see her goals realized peacefully. Maybe they wouldn't have changed peacefully, but we don't know, because she didn't want to even try being peaceful in her approach.