We just need to fine copyright holders who abuse the system. Currently they are free to file as many false DMCA notices as they want, with zero drawbacks or consequences or costs for the filer and the burgen of proof on their victims.
If every false DMCA notice resulted in a fine for copyright holder, or even better, filing X false DMCA notices resulted in a copyright holder's ban from filing more DMCA notices, then I think the problem would resolve itself in short order.
I have heard that some folks use DMCA as, basically, a "protection racket." Not sure why they would do that, because there are probably easier (and less risky) ways.
But, as long as there's no penalty for filing a false DMCA claim, there's nothing stopping the "blunderbuss approach," which is hugely popular, with lawyers.
It seems to me like a reasonable starting point for a solution is to allow hosts to bill claimants for "reasonable costs" (I assume that's legally well defined) for false claims.
That provides an incentive for companies to respond to the screaming demand from their users to have better review.