So person C enters the relationship and they form a polyamorous-trio, but instead of it being a true trio, it’s more like A & B still have their relationship (now burdened) and A & C have a relationship, but B & C don’t engage much. This is the exact scenario I have witnessed in the only 2 households I’ve ever known doing it.

That is in fact common, but would also not result in "moving in" or "forming a polyamorous trio". That's exactly not the point, it's just one person having two relationships and - hopefully - each of the partners is fine with not having 100% of their partner. Which many people actively enjoy mind you, not spending all the time sitting on top of one another.

In fact I would say that from all the poly couples I've know over the years, very few are trouples and want to move in together.