The recommended solution for embedded content is the Storage Access API, which has broad browser support. (I don't know that YouTube has implemented it yet, but I'd be surprised if it's not there soon.)
Oh, that's worrisome. I'm trying to implement it in our embedded content soon, because some of our customers got caught up in Chrome's third party cookie blocking 1% trial that started a week or two ago.
Any specific problems you ran into that you could share?
From Google's own page: "Work is continuing to resolve all remaining blocking issues, before standardizing the API."