If youtube won't host the information, it becomes a search problem: who has voted on this link? How have they voted?
Certainly the easiest path would be to create a new centralized site that hosts all these votes. Then create an extension to add these downvotes back in.
Ideally you'd probably want to also have folks be willing to track views too, since downvotes is pretty much a consideration of ratios; 22 downvotes doesn't sound like a lot if there's a million views but it's a lot if there's 100 views.
So now we have an extension that tracks every youtube video you view and submits it to some centralized site, along with upvote/downvotes.
It'd be excellent to try to evolve a more distributed p2p architecture for doing this, but short of recording it on a blockchain- and a roll-up side-chain probably won't help- there's not a lot of good ways to do this. An intermediary step would be to have folks create micro-pages that record their votes. Then we can search "link:https://youtube.com/1a2s3d4f" to use google to go find all the votes people have created.
It's sad & bitter Youtube is de-democratizing this way, is serving their customers & not the audience, but it does interestingly highlight what a hard problem it is to create real demoncratic systems online.
It would be better to get a count and ratio of like to dislikes but Google doesn't want that so comments it is.
The other option is an off-site like/dislike page that is hosted off of Google's servers and not subject to their tos. Maybe accessible through an extension. Abandon YouTube like/comments since they clearly can't handle the engagement and host the discussion on a third party page - one page with a like/dislike count for each YouTube video and with better forum type tools than YouTube comments offer. A good opportunity for a third party to build a tool that solves the problem outside of YouTube.
I'm guessing YouTube regrets putting comments and votes on its videos in the first place. Embedding YT videos still gives YT views (unlike, say, reposting a gif to 4chan) and I would think that they'd prefer FB/Reddit would just eat the controversy over curating comments and votes instead of having to subsidize it themselves.
Last time I tried to switch to an alternative, I ended up with my messaging spread out to many apps. So I ended up going back to the common denominator: Whatsapp