Or something less drastic: Someone is knocking in your front door. You open it and see... Who is it? How do you actually rewrite this branching point as protagonist's decision?
If were me...
If I'm positing this question in that manner, it means that SOMEONE in the story is asking it. The answer is not necessarily the truth, but in the three perspectives it's still possible to give agency to the protagonist if you're willing to assume that the protagonist is not bound by the truth/Canon that has been established.
1st person: The main character can lie. If we hear the thoughts, there's possibly a reason for it. Conflicts between this and past established moments can be reasoned as lies or seeing things incorrectly. Memory is fallible, etc. Might even be that the main character doesn't recognize the identity established or disagrees. The main character's perspective can disagree with the facts established, therefore still retaining agency and choice.
2nd Person: If I'm telling in second person, it means that I've already granted a great deal of agency to the reader. As such, putting thoughts into the players head is depriving them of their own agency, and breaks immersion heavily if they are constantly making choices that relate to the "Reality" of the story, or reminders of a game. A question like "Who do you see?" Is very open ended, and encourages involvement in an audience that must wait on a delay in response at best, OR write it themselves. Both of which encourage creativity, but also require WORK, which is the opposite of the goal in having a story told in the first place. Short Version: Giving a question like "Who is it?" In second person immediately draws attention to the fact that "this is a GAME" and as such set certain expectations of control and directorial ability you don't find in other fiction. A BAD choice in this type of story can RUIN immersion and cause incredibly angry pushback. (For a RL example, see the Ending of Mass Effect 3)
Third Person: This pushes back the involvement, making the characters more independent from the reader. As such, a "Who is it?" question addresses someone not personally involved in the decision, and thus already has made the question less personal/meaningful. A question like this is less immersion breaking, because the player sees the protagonist as separate from themselves completely. As such, the question is the equivalent of another toy being brought in, and the selection more of a "A New Challenger Approaches" type of decision. There's fewer stakes on the decision, and if a poor choice is selected, it's an easy matter to go back at that choice and go with a different selection. The question is additive, expanding the scope of the story, and bringing in more, rather than demanding the player provide input.
Phew... So... there's a LONG answer to a very short question. I think it's possible to write from that question and still have agency, but it's better to think about how it's going to be seen by the reader first, and how that decision is going to affect their experience of the story. If somebody wants to know more about a certain point in the story, for example, the CHYOA format allows you to go back in, write a fleshing out/flashback or side path in response to a need, and then retain the story or change it based off of what you've done in the past. Need a REASON for the main character to betray their friends and not make it seem out of nowhere like hollywood? Write a quick flashback in the story where you can see WHY they turned against their childhood friend.
Of course all of this requires the dreaded W word. WORK. I think you can make anything work... as long as you're willing to work hard enough and take enough time with it. I could be wrong of course, but I did take a long time and had a lot of fun writing this. So far as I'm concerned... this was worth it even if you disagree with me because it served as a nice place to get my own thoughts down on this. Thanks!