Yeah, I think the chief one against is that the pair's barycenter is below the surface of the Earth. But if the moon were a third again further away that wouldn't be the case any more. The Moon is receding, but I think it'd take way to long to get that far at 3.8 cm/y... so... 33-ish billion years until you could call it a planet? I dunno I'm not even bothering with a napkin here. Anyway it's kind of a moot point unless you want to Kerbal something together to give the thing a shove.)
I guess it mostly comes down to even the IAU definition of "planet" being necessarily broad and seemingly arbitrary, just because it's a pretty old word that don't work so good no more. It's basically an attempt at technical a definition that has to be jiggered to mean "bright thing in the sky what moves around, but not the sun, moon, and stars, or any weird shit my primitive ass doesn't know about yet". That's a weird club that includes such disparate objects as Jupiter and Mercury, but leaves out dwarf planets, Kuiper objects, the moon, and probably a bunch of other shit that has a hell of a lot more in common with Mercury than Mercury does with Jupiter. And that's fine, because anybody who needs to be told that Jupiter is a planet probably doesn't care that it's a class I gas giant or that its magnetosphere is fucking crazy and that it really doesn't in any way resemble Mercury other than that it's also mostly round, bigger than a bread box, made of stuff, and orbits around something else. That describes most of the things in the universe that could be described as things at all. Kind of like how we put "fish" in the names of things that clearly (or not so clearly) aren't fish, but they're in the water and they're fishy so fuck you. Way later, after we figured out what clades most of those critters belonged to it became really convoluted coming up with a definition of "fish" to include all the things you'd normally think of as "fish," without roping in a whole mess of other dumb things. But hey, like "planets," it gets across what you mean most of the time without splitting too many hairs, and fish are a lot more likely to be relevant in some way to your day-to-day existence.