> This probably can’t actually work in practice. Why?
It often does. Because of the asymmetric penalties in the DMCA. Yes, they are required to process all "properly submitted DMCA notices", which means they can legally ignore "improper" notices. But to determine "properness" requires a lot of work (read as: costly) and if wrong in that determination, they become liable for damages (which for copyright can be statutory damages of $150000 per violation). Which can make being wrong in the "properness" determination very costly.
On the other hand, they can "take down" the content, even if the notice is improper, and there is no penalty for doing so. This path is easy (read as: very inexpensive).
So most hosts simply default to the cheap alternative. Take the content down, and let the user who posted the content file a counter claim. This shifts the liability from the host to the user, thereby allowing the host to maintain their DMCA safe harbor and avoid those potential statutory copyright damages that /could/ result from the alternative path.
The DMCA does include a perjury penalty against the original submitter, but in order to enforce that the receiver has to separately file a lawsuit against the sender alleging they perjured themselves in creating the notice. This, also, is a very expensive path to take, and so third party intermediaries are not likely to pay this amount of cost on behalf of any given small user. And unless the person who posted the content originally has sufficient personal fortune to pay for this path, it is almost never taken, so senders of false DMCA notices never see any negative feedback on their activities.