I think a DM both is and is not a player, depending on which meaning of player is being used at the time in the discussion. It's like the term "brother"- generally, we can all agree that if I share the same birth mother and father as a guy, he is my brother. Most (but not all) would also agree that a guy adopted by my parents counts.
But there is a spectrum here, which depends partly on the observer. What if we share one parent? What if the person assigned male at birth is now a trans woman, or conversely, if they were assigned female and are now a trans man? What about a close friend who I call "brother" but to whom I am not related (except inasmuch as we're all distant cousins)? What about my parents' male dog, who they treat as one of their children?
Likewise, is the referee of a basketball or soccer game a player? Is a person who writes adventures a player of those adventures? Does it matter if there is a procedure for writing those adventures?
I'm not sure that most people would say yes. I acknowledge that a GM/DM is a different sort of referee, a much more active participant in the game than an adventure writer or football referee, but I'm wondering how far the notion goes, and, like you, how useful it is. Different rpgs have different degrees of DM freedom, but generally, they all- as far as I know- put a far larger portion of the responsibility for there actually being a game on the DM. Generally, an rpg can go on if one player is missing, but not if the GM is missing. Generally, the GM does far more work to make the game actually happen. If I am running a campaign, it is my game, nobody else's. Without me that game doesn't happen. Whereas if Bob or Dan are sick and can't come, but we still have a quorum, the game can go on (although some groups cancel if not everyone can make it). I see the difference as pretty fundamental.