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.