I am also experiencing this issue, and I have found a temporary workarounds that works for me.
Option 1: Enable autoplay in your browser settings. Go to Settings > Site settings > Autoplay (bottom section) and enable autoplay. Then, restart your Brave browser.
Option 2: (Without enabling autoplay) When you open Brave, open and use a new tab instead of using the last session tab that Brave loads.
My screen doesn’t turn off when I do one of these option. I think by default autoplay is enabled that’s why not everyone is having this issue. And I feel like this happens when the last session tab is loaded from opening brave it is like in background state and stays like that even when playing video since it keeps playing even when the screen turns off, but if you use a new tab or if autoplay is enable it works like normal.
I also tried Beta and Nightly and both is having the same issue for me if autoplay is turned off. I use the same workarounds and it also worked for both versions.
I don’t know if these will work for others too. I am on Android 14