Step one here is to try to separate (a) browser functionality breakage, which can be fixed by changing the browser, from (b) youtube refusing to serve certain content or certain pages or certain resources to Tor users.
That second one has been an issue lately -- youtube thinks that users from the Tor network are crawling it too much, or asking it hard questions, or whatever they're doing, and it responds with 403 answers to some questions, which will definitely break functionality.
Obviously one solution, is to allow canvas on YT: but that defeats the purpose - and it wouldn't surprise me that the reason for doing this is to force users to remove privacy protections. The fact that canvas RFP protection is easily detectable can be used against users
Another solution is to play whack-a-mole with JS/elements: and that's not a long term solution either
Tom's ticket on poisoning canvas is probably the ultimate answer