Well it's been a long time since I've picked up my s3 so I can't speak on the ROM but I did have a couple apps to recommend that may save you some headache. First there Google Cast (
https://play.google.com/store/apps/details?id=com.google.android.apps.chromecast.app) which is free and it has a cast screen option which worked wonders for me back in the day. Played many of movies through it to my TVs. Then there's the paid app Allcast (
https://play.google.com/store/apps/details?id=com.koushikdutta.cast). There is a trial period where you can try it first but this is my go to app now. Sorry I can't help you on the built in features that aren't working anymore but maybe you'll like one of these alternatives. Best of luck to you
PS, this was the best and only ROM I ever ran on my s3 so nice choice. Still following these guys and their development. They're great.
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