Yeah, abusing the DMCA because it's the only tool you can think of, that's a great idea.

Come to think of it, our society grants search engines the privilege of keeping copies of copyrighted material in exchange for the services they provide.

It might not be unreasonable to say that this privilege comes with a certain responsibility to ensure that those copies do not cause excessive harm to others.

So although Cloudflare is the one that fucked up, Google et al. also have a responsibility to do whatever they can to protect the public. They should do what they can, with or without Cloudflare's cooperation.

I don't believe so. Cloudflare had this responsibility and they messed up. They are the only ones liable.

Cloudflare cannot delete documents from other company's caches. The actual deletion must be performed by Google and Bing, who can then sue Cloudflare for cleanup costs if the latter is unwilling to cooperate.

When there's an oil spill, we don't wait for the oil company to come and clean up their own mess. Others clean it up a.s.a.p. and (ideally) then make the oil company pay the fines and damages. CloudBleed is a virtual oil spill. They literally sprayed other people's private data all over the internet.

Yep, it's a Section 230 for the search engine, probably -- the unfortunate data came from Cloudflare.

Why can't they just search for pages with data after the closing HTML tag

That isn't uncommon enough. Searching for one of the CF- http headers in the source code, however would work.