Tbf it was not that simple either. Most Catholics did not really have issue with Orthodox Christians, it was largely ecumenical and political matter. First crusade started when Byzantines asked Pope for help. And ofc rebelling troops that would later sack Rome were mostly catholics themselves iirc. Things only escalated over time, and even then catholics and orthodox Christians mostly lived peacefully and fought wars because they were bored (obviously not literally). When i say most of people, i mean average citizens and not clergy or ruling classes etc. And ofc they did not always live peacefully.Yeah, that is bollocks because the Catholics hated the Byzantines until the later were so weakened they decided to jump in. At that point Jerusalem had been under Muslim rules for a centuries and none of "Europe" gave a rat's ass.
Constantinople was Christian when the crusaders took it "back", destroyed the Roman Empire, established puppet states and left the rump region of East Rome in chaos.
But point is that they mostly fought because everyone fought everyone. When Ottoman puppets (which would later become Romania) revolted to gain independence they were supported by Austrians, and later they fought each other and Austrians as well.
Which all just proves that anyone can fight against anyone. Closer to this time period as mentioned there are Varangian guards, and Crusaders had so called Turcopoles. Other side opposing crusaders had Christian troops fighting for them.
And Christian church ofc used to be pacifist, yet they would launch wars motivated by the same religion.
But it definetly would be nice if conquered areas would have little more revolts which could be supported by factions of same culture, and if revolt is successful and there is no faction that culture used to belong to, that faction could be re-established.