Even youtube's app itself doesn't allow that unless you pay. I suspect they've nobbled most browsers into not allowing it, either by technical measures or (more likely) the strong-arm tactic of saying “if you don't block this we'll find a way to make the entire of youtube practically unusable on your browser”.

I've been using Grayjay recently which does allow that, amongst a number of other useful features (integrating other media sources, lack of adverts every few minutes in some content). Might be worth considering as an option.

Kagi’s Orion browser on iOS is able to play YT vids in the background.