Disclaimer: I think it's obviously true that at least some conflicts can be considered to have been caused by religion, my issue is more in how common and how big those conflicts are.

So there are two problems here: First is the fact that these hypotheticals for the most part can't be untangled with any degree of objectivity. Second is the question of how to evaluate religion's impact on direction versus magnitude. Which is to say, it's entirely possible that you have the conditions for a war that simply need something to shape them into a coherent conflict. Your example is actually pretty illustrative here, so lemme just:

I think if you look at least of a few of the European Crusades into Holy Land, you'll see religion meets this standard.

The problem here is that religion was so tied up in the culture and institutions of the time that it's nearly impossible to envision a Medieval Europe without religion (see first problem). The thing about the Crusades is that they were directed by the Church, which was eager to assert its influence, and you don't even have the Church without religion. But then you don't get nothing, because the other thing the Church was doing was trying to reduce conflict between Christians, so the Crusades were essentially the Church redirecting this violence externally; without that you don't get less conflict, just different conflict (see second problem). Historically the lack of overarching authority to direct violence tends to lead to more rather than less violence (essentially a corollary of the square-cube law), so "just get rid of the authority holding a notoriously fragmented and warlike region together" isn't exactly a recipe for less conflict. If we instead imagine that the authority holding Western Europe together was secular (as much as anything can be secular in this period), then the number of warlike fragmented regions that didn't expand violently after being politically unified throughout history is indistinguishable from zero. The analogues here would be the Italians (under Rome) or Mongols, not exactly the most peaceful peoples.

So to get back to your heuristic, the Church undeniable caused the Crusades, but it'd be a much harder sell to say that without the Crusades there wouldn't be something else in its place.