The question goes even deeper than simply asking who the protagonist is; it makes the designer have to decide how much control is handed over to the player and of what kind. Does the player simply write the story, controlling the decisions and actions of all the characters, or do they only act as one causal agent? And if the player has reign over the main protagonist, how much room does that provide the designer to define the character at all? Does the designer create a doll house or a novel? Would creating a sandbox for role-playing truly create the best experience?
I have my doubts.
Frankly put, outside of total simulation I do not find the idea of handing over the entire character to the player appealing for narrative purposes. The best example of this happening in video games can be seen in western role-playing games. For instance, at the beginning of an Elder Scrolls game the player creates an entire character from scratch which is then dropped into the role of hero. While it works (even though most people went through Skyrim guiding one of the most contradictory characters one could possibly imagine, it was not so much an issue that it detracted from the overall experience), I would say that it fails to create the proper amount of drama and narrative magic to really be seen as a reasonable approach unless the player is of the minority who sticks to a strict regime of role-playing consistency. The writer will always be better than the player when it comes to creating characters that fit appropriately in the fiction.
So, if the writer holds the power to creating the protagonist, is the player left to having no agency and being filtered through a linear plot that takes little advantage of the properties games have to offer? Of course not!
This is a problem I want to solve. Does the player determine the decisions of the protagonist, or does the personality of the protagonist determine those decisions? Which answer conveys more character? A player given the goal of obtaining victory and optimizing their path towards such will not make dramatic decisions in these situations, especially since they are so disconnected from the emotion of the fictional situation. A character's personality will lead to that character making particular decisions that in turn affect the game.
Essentially, my answer is that the player does not enjoy themselves playing the the part of the character, but rather the character and the game are two separate entities, both dynamic and prone to change, and in constant influential relation to each other. Some titles have already started to explore this idea, but it needs to be developed further and stripped of the violence of current games.