"No, because by putting your content back up the host is opening themselves to liability if it turns out you are wrong. Nobody is going to do that"
No, the point of DMCA is the provider is agnostic to who owns the content. Its the making a decision on who owns the copyright that opens them up to litigation, because they would lose safe harbor protection. Incidentally, NOT putting the content back up would be a violation of the DMCA, which also opens you up to litigation.
"The person who filed the complaint is a mailinator address, or in Russia/Iran/Cuba or works for a corporation with more lawyers than you."
So they're either a 1) liar, in which case your content is back online and you're back where you started or 2) a big corp who can threaten to get their way? Welcome to the fundamental problem with the DMCA: it can be abused.