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No. What you describe is correct: No background play via the yt app unless you pay.

Yes YouTube Premium will play with the screen off (using the app. No idea about using a browser).

In a browser, it works even without Youtube Premium :-)

Firefox mobile, m.youtube.com, "Video Background Play Fix" browser extension.

Works for me without an extension, you just need to click play again after leaving the YouTube tab/locking the phone

So does ReVanced YouTube

On iOS I use Brave and it works fine.