Earlier quoted context omitted.
You've got it backwards. It's on the defendant to prove that their use is fair. The plaintiff has to prove that they actually own the copyright, and that it covers the work they're claiming was infringed, and may try to refute any fair-use arguments the defense raises, but if the defense doesn't raise any then the use won't be found fair.
It's true that the process is copyright strike/lawsuit -> appeal, but like I said, it's in their best interests to just prove that it's fair use because otherwise the judge might not properly consider all facts, only hear one side of the story and thus make a bad judgement about whether or not it is fair use. If anything, I'm just being pedantic, but we do ultimately agree here I think.
That said, I think we do agree. The plaintiff should be prepared to refute a fair-use argument raised by the defendant. I'm just noting that the refutation doesn't need to be part of the initial filing, it gets presented at trial, after discovery, and only if the defendant presents a fair-use defense. So they don't have to prove it's not fair use to win in every case. I'm probably also being excessively pedantic!