When developing a WebXR app, it should be not necessary to regard browser specific implementation details (that’s why you have a standard). According to the official documentation, Oculus Browser supports WebXR since version 7.0. If you encounter issues with the WebXR implementation, you can file bugs right here:
https://developer.oculus.com/bugs/
I’ve tested the official three.js VR examples on a Oculus Quest (with Oculus Browser) some time ago and they all work.