This app was mentioned in 3 comments, with an average of 1.67 upvotes
OK, so I came across this app that looks like it does exactly what I was looking for: https://play.google.com/store/apps/details?id=com.ekm.youtubevr3dvideos
It did require a $0.99 purchase to unlock search functionality, but I'm fine with that. I'll report back if it sucks or ends up not being what I was hoping.
EDIT: So this seems to work fine but the way it works is that it opens two instances of the YouTube video and attempts to sync them up. Sometimes each side can get a few frames out of sync (which, to its credit, it resolves pretty quickly), but it also doubles the audio, so there's a very noticeable echo.
I think what I'm gonna do is use a third-party app to download longer YouTube videos that I wanna watch, and then use this to watch them. An inelegant solution for now. Maybe I'll end up writing my own app :P
Would this app be useful?
https://play.google.com/store/apps/details?id=com.ekm.youtubevr3dvideos