We have always had a smallish group. We have always all wanted to play.
taking turns DMing the same party means we often had DM PCs that later were just PCs.
I did this once in 5e; I often forgot special abilities and spells due to focusing on the game and players—-opposite of hogging the spotlight and favoring self.
I think this notion that DM PCs are there for the DM to grandstand and hog things is a gross overgeneralization.
it really comes down to who is playing. As a DM I am cringing if players have poor luck. I am rooting for them and would not take choice items or make a power play for a pet npc.
that said if you ride with outlaws, you hang with outlaws. If the party is ok with killing one another, the dm PC should not be off limits.
Yeah, I think that's fair.
To my mind, there's a difference between "the DM's PC" and a DMPC. The first is what you describe: the DM is also a player with a character. Some groups might have trouble juggling that, but I know it works fine for others.
A DMPC, on the other hand, is something like "the DMs favoritest PC 4evar!" The character is often overpowered, and might be a recurring PC in that same setting or even in others. Adventures often end up revolving around the DMPC, too, consistently and in ways that go beyond the NPC as plot device (who does their important thing, but then leaves the spotlight).
Back in the 2e era, I played for a while in a long-running campaign with a DMPC in this sense of the term. (There were 6 players, so it wasn't necessary to fill a spot in the party or anything.) In this game, the DMPC was the object of every quest, the key figure in every storyline, a buddy of godlings and kings, and powerful beyond their stated level. While there were a couple longterm PCs who had prominent roles in the story, most of the PCs were effectively supporting cast in a grand multicosmos-spanning story about the DMPC. I mean, the story was vaguely interesting, but I mainly played just to hang out with friends; and when I left the game, I didn't miss it. Or even remember much of it.
So that's my DMPC story, and why I tend to avoid them. I imagine it largely comes down to one's personal experience dealing with them.