I have 3 headphones that are type-c, Bowers & Wilkins PX7, Sennheiser Momentum 3, and Google Pixel USB-C earbuds.
PX7 and Momentum both work when wired usb-c to usb-c on my pixel 4 xl and pixel slate, but do not work on any of my stadia controllers. However the Google Pixel USB-C earbuds work on any platform.
is the Stadia USB-C limited to Google Pixel USB-C earbuds ?
No it's not limited to only Pixel Buds, I've been able to successfully use some Jabra USB-C headsets with my Stadia controller when I first tested it. That said, I suppose there are some that don't work out there that you've discovered.