... does not depend on your assumptions about how players do, would, or should behave. It is simply a property of the game, i.e. (among other things) of how players *can* behave. If the responder (for whatever reasons) throws a coin in case of indifference, then there still *exists* a SPE, it is just not played.
If you want to arrive at a game without a SPE, you have to change the rules of the Ultimatum game, e.g. by demanding that the responder *has to* mix (with strictly positive probabilities) in case of a zero offer. However, this is a different game, and also the restricting assumption would be rather ad hoc.