Yeah I mean, it's not well written, not going to lie. I mean you can see what Ubisoft was trying to set up, but, the execution of the plot, and motivations, doesn't carry well through each game, because they didn't put much effort in it. They just used it as a thin framework to justify bouncing around history for Ancient Assassin Simulator in historical locations, the franchise. Which is what really bothers me personally, because I can see how cool it COULD be, but the writing is...not up to it.
I've always kind of lumped their motivations into something similar to Jedi/Sith, in that yeah, there are other bad things the Jedi could be dealing with in the galactic republic, but they simply can't put out EVERY fire that exists, so they focus (mostly) on trying to counter the supernatural machinations of the Sith, as they are the group, most likely to stop them. To me, the whole Progenitor conflict, between the Order/Chaos factions, fought out via a proxy war between the Templar/Assassins, is similar to that. But it's just not written well. So it falls flat. Which again, annoys the shit out of me, because the entire framework is built around "genetic memory", which, for me, as a Dune fan, is like catnip for crazy concepts and stories that could be told. But they never really do anything with that, other than to justify the era hopping.