You aren’t someone when playing a video game besides yourself. A third person view doesn’t suddenly make people unable to feel as if they’re playing as that character any more than a first person view does. For example, people can have a similar feeling even from books, with no agency.

You’re making a weird argument based on some purity metric. Either way, you’re playing a video game and controlling a character in the game. Neither view let’s you be that character. Both let you be immersed and inhabit their role in the world.