Not being able to play Youtube in the background on your phone is unfortunately one of the main appeals of Premium. There's a lot of good mixes, concerts, etc that I play for the audio while doing something else that I can't do without Premium unless I wanted to leave my phone unlocked (and pray I don't pocket click a link).

My iPhone running Safari and uBlock Origin lite is able to do this. I don't have the youtube app installed. I don't even think the ad blocker is necessary for background audio, but I don't want to see ads.

1. Go to youtube.com in the browser, play the video, switch back to the home screen. Video playback will stop, which is a good default behavior.

2. Swipe down from the top of the screen which brings up "Notification Center" which somewhat strangely contains a playback control for the browser.

3. Press play. Audio resumes. If it's part of a playlist, you don't have to manually advance, it will play automatically.

No ads, no youtube premium subscription, no "desktop mode", no sideloading, no additional apps other than the beloved ad blocker.

I'm fairly certain if you use a browser and the desktop version of the site you can listen with the screen off/locked.

Firefox Android can play audio even when the phone is locked, and I use it regularly.

If you're on Android, YouTube Revanced does this (+many other premium features)

You can open youtube in the phone's web browser and install an extension that blocks a site's ability to tell when focus has left the page/app. This is how I listen to some music on my phone while working out.

Ad blockers help with the constant nagging about "open in the app!"

NewPipe