The Rams care. They are not going to move a player of his caliber somewhere he isn't good with. Also doing that opens up him saying fuck you I'm retiring. And they can call his bluff, but that eats into his value in trade. So bottom line is when this situation happens you put it out there that you expect a top pick, which the Rams reportedly have. Then you let the QB find a spot he likes where the pay is there too. Then you work the deal.
Now whatever team sells Stafford on his future with them will have already known the rough cost of the trade as well, due to the cost having already been discussed. Also they can concurrently discuss these details with the team as they proceed with Stafford. Comms are all connected on this stuff.