What makes something an "outcome" in Savage's theorem is simply that it follows a certain set of rules and relationships - the interpretation into the real world is left to the reader.

It's totally possible to regard the state of the entire universe as the "outcome" - in that case, the things that corresponds to the "actions" (the thing that the agent chooses between to get different "outcomes") are actually the strategies that the agent could follow. And the thing that the agent always acts as if it has probabilities over are the "events," which are the things outside the agent's control that determine the mapping from "actions" to "outcomes," and given this interpretation the day does not fulfill such a role - only the coin.

So in that sense, you're totally right. But this interpretation isn't unique.

It's also a valid interpretation to have the "outcome" be whether Sleeping Beauty wins, loses, or doesn't take an individual bet about what day it is (there is a preference ordering over these things), the "action" being accepting or rejecting the bet, and the "event" being which day it is (the outcome is a function of the chosen action and the event).

Here's the point: for all valid interpretations, a consistent Sleeping Beauty will act as if she has probabilities over the events. That's what makes Savage's theorem a theorem. What day it is is an event in a valid interpretation, therefore Sleeping Beauty acts as if it has a probability.

Side note: It is possible to make what day it is a non-"event," at least in the Savage sense. You just have to force the "outcomes" to be the outcome of a strategy. Suppose Sleeping Beauty instead just had to choose A or B on each day, and only gets a reward if her choices are AB or BA, but not AA or BB (or any case where the reward tensor is not a tensor sum of rewards for individual days). To play this game well, Savage's theorem does not say you have to act like you assign a probability to what day it is. The canonical example of this problem in anthropics is the absent-minded driver problem - compared to Sleeping Beauty, it is strictly trickier to talk about whether the absent-minded driver should have a probability that they're at the first intersection - argument in favor have to either resort to Cox's theorem (which I find more confusing), or engage in contortions about games that counterfactually could be constructed.