Utilitarianism is normative, so it means that your utility should be the average of the utility of all beings capable of experiencing it, regardless of whether your utility currently is that. If it becomes a weighted average, it ceases to be utilitarianism, because it involves considerations other than the maximization of utility.

one tiny change in the mind would have to suddenly shift us from fully caring about a mind to not caring about it at all, which doesn't seem to be what humans do

Consider how much people care about the living compared to the dead. I think that's a counterexample to your claim.