I do intend making GM a player to imply that players may become more like GMs. Once a GM is a player, then it is the case that GM powers are held by players.
I ... don't find this to be a useful way to discuss things, for the simple fact that the idea that there is a long-standing practice of having a "GM role" that dates back to the first RPGs, and has continued ever since.
At the beginning, there was the "referee." Regardless of the various shifts of nomenclature since then (referee, DM, GM, etc.), the idea that there was a specific role that was differentiated from the players has continued in almost all games, to the extent that we can actually describe certain games that
do not have that role as a specific (and very small) subset of RPGs. Fiasco would be an example of a game without a GM.
To use the common analogy, a game of basketball has a referee. If you were to play an informal game of pickup, you might say that all the players are empowered to make calls (subject to the other players), but you wouldn't say that the players are referees.
For this reason, it is more useful to think of two different roles in RPGs- GM and player. However, different games might have different
divisions of authority. The usual debate (which is a recurring, and tiring debate) is over the amount of
authority over the fiction that these roles might have. There are other debates as well (the amount of authority over the rules, or the amount of authority over the players' declarations), but it's all about the division of authority inherent in the role.
In the most traditional RPG model, the GM has sole authority over the rules and the fiction, while the players have sole authority over the players' declarations. Other models have different (often shared) authority over the fiction and/or the rules.
I don't think it is helpful to confuse the issue to say that
the GM is a player.