Well, I'll be. I google windows version 2004 and wireless 360 adapter and found a youtuber that had already put out a fix:
https://www.youtube.com/watch?v=SNX2GDzfT6A
It seemed to work without a hitch, but I did it remotely and won't know until I get home (but the device is recognized now)
Basically, it looks like MS took out support for the third party adapter (but you can edit the Driver ini file and bring it back)
I found this yesterday too, this works. Be sure to switch out of test mode after you've installed the driver and confirmed it's working.
Long story short, like you said, Microsoft removed the hardware ID for the Chinese made model (0291) and only left the legitimate model (0719). You won't be seeing a hotfix from Microsoft for this, so modifying the driver is going to be the way.