Facing the same thing too! If my baby doesnt burp after few mins of patting, i will just put him down, esp if he is already asleep. If he were to cry in the middle of his sleep, i will pick him up to burp, which he will sometimes. And yes, my baby farts a lot as well. I guess if they dont burp, they will pass gas instead.
Actually all that you’re doing is okay. Not all babies need to be burped so much if they’re not so gassy and they don’t vomit. And if she burps on her own, then that’s great. Let her sleep:)