So I solved the issue on my own before having to contact Apple support:

For some reason, the Apple TV had no "default user" set, so everyone in the family, myself included, was listed as "additional users". I don't understand why the Apple TV doesn't let you "select" a default user from among the additional users. Seems like a design flaw to me. But it means that what I had to do was delete myself as an "additional" user, then re-add me as the "default" user.

After I did that, the Apple TV asked me which Room it should be added to, and everything started working just fine.

¯\_(ツ)_/¯

Grey