So firefox can't do much about it without actively trying to circumvent YT and YT specifically.
I don't think browsers made the turn you mention. It's more like browsers became more and more capable and web developers made use of it. Sometimes it's annoying because most websites are not websites anymore but apps (GUIs) that run in the browser and some of the web sites/apps people use could never work without it. Sure, we could all deploy those apps onto our machines (or have them deploy automatically in a sandbox) and there were actually technologies that did just that (think java web start or whatever the name ended up being) but they lost to what we have now: running these apps in the browser.
Also, you can't have an ad-free experience if the price of using a service is that the ad is delivered to you. On YT you can buy a subscription and you'll see no ads. But sure, most sites don't offer this.