As for the question and the poll, to have a name change happen there has to be a reason, in the case of the Titans it was a move and the owners desire to have a name fitting the city which is called the "Athens of the South" and was not done under duress to my knowledge.
The Browns was a hostile claim of name which if that were to happen it wouldn't seem to me that anyone really has a claim considering that Cleveland has a team and covets the Browns name , LA got the team from another city ,lost it to St.Louis ,so if the name changed there would seem to me to have to be another city involved. because the request for the name change would have to come from LA and 20? years later ?
Gotta believe there won't be a name change but if it was to change it'd seem to me only in St.Lou so under those assumptions,I'd say yes I'd still root for them, but Stan would need a reason which I can't see because no one can assert a superior claim on the name and the league isn't going to expand ,the mathematical symmetry they have now suits them too well and the Rams name would have to be then for a team relocating to LA,so who among the prospective LA teams would want to change their name ,surely not the Raiders nor the Chargers ,maybe the Jags, but HOW could they assert a claim over current owners of the name ?