Eventually, and over a longish time that got whittled down to one (“monotheistic”) but that hankering for “other gods” wouldn’t go away, so now it’s THREE, except it’s one…

I have always found the concept of the Trinity to be confusing, and I can't say that I fully understand the reasoning for it. But I don't think it stems from a "hankering" for other gods. Rather, I always thought of it as an effort to solve a fundamental conundrum of the New Testament. Namely, that Jesus (and the Holy Spirit) are certainly talked about as being different persons from God but are themselves divine. IOW, the Trinity isn't an effort to satisfy a need to have a separate "God of X" that's different than the main God, but rather trying to reconcile the divinity of Jesus with the fact that he is portrayed in the Bible as a person who isn't the same as God.

Again, not a comparative religion expert, but I agree that the saints seem more the concept that fills the other-god-hankering role more than the Trinity, and they certainly played that role - especially in syncretic religions that blended Catholicism with local pre-existing beliefs.