I don't think OBJ is responsible for the delay in his signing. IMO the Rams want to know when OBJ will be ready to play before they sign him and that's a big unknown at that point. I never thought he would sign before October anyway so I don't see any drama at all in the fact that he's still a FA.
It could well play out as you have suggested - that OBJ gets an offer he can't refuse elsewhere (much like Miller did). But I don't see any reason to think that OBJ remains unsigned simply because he's asking for too much money.
As far as OBJ only being able to play a half a season (or maybe even less) I think OBJ would be inking a 3-year deal if he does sign with the Rams. So from that perspective the Rams might be satisfied getting whatever they can get from OBJ for some of the 2022 regular season plus playoffs, then build on that over the next couple years.
AlbaNY_Ram