I think throwing in wish fulfillment is unhelpful in explaining the immersive play style. There are certainly styles of play where wish fulfillment can play a role, but I find that is a separate thing from say a person simply wanting to have complete control of their characters thoughts and actions. I do think there are very different approaches here. Some people want characters who are effectively extensions of themselves. They might want to be constrained by POV in the setting, and they might want the character to be different from them in certain ways, but I think a lot of people ar looking for the experience of being in the world. But there are also others who want to inhabit a character, and they develop character immersively, rather than through system. And there are people who want characterization through mechanics. There is nothing wrong with any of these approaches. The problem is when we assign value to one over the other, or use language suggest some are more remedial, less sophisticated, childlike or an attempt to fill some lack in life (anyone spending a large amount of time playing games, can be accused of that).
I do think there are games that engage with power fantasy. But I think that can happen in any of the styles we are talking about. And there isn't anything wrong with it. Some folks blow off steam by trying to become king in a fictional world or video game