"Youtube requires a browser that supports Web Environment Integrity in order to provide you with the best quality in streaming content. Please upgrade to a browser supprting Web Environment Integrity. Chrome <link to chrome> and Edge <link to edge> are our recommended browsers"
Or words to that effect. And it'll likely gradually become the same for most advert funded sites.
EDIT: Actually I don't think this will become a thing until it is hitting their bottom line. That is if enough people switch away from chromium based browsers. Just like it did for ad blockers.