Currently, OBS uses the system of "stream keys", which is the basis of how most stream services understand authentication. The beauty of this system is that it's easy to configure, and in most cases is a "set and forget".
Facebook uses this system a little differently than most, where it requires a different stream key for each event. While it's not the method most streaming services use, it does help in ensuring you're streaming to the correct "channel".
The way GeForce Experience does it, is that it uses a separate method of authentication (ie. you log into Facebook via GeForce Experience), and then it either fetches/creates a new stream key and uses that automatically. This hides the hard work from you, which overall provides a more streamlined experience.
As it currently stands, we are working on doing something similar - that is, via OBS, you sign in to the streaming service, and we do the stream key fetching/generation in the background. There is no ETA for this feature, but it's definitely in the works.
Note: In terms of your phone, it works instantly because it's the Facebook app and Facebook also hide the hard work from you. They make it harder for other services for a reason.
