If there would be a button for claiming copyright that has a checkbox above saying "I accept liability for falsely claiming copyright." the problem might fix itself after enough lawsuits?
A similar power imbalance exists between workers and employers. If the worker doesn't get paid, they face the prospect of having no money for rent, let alone suing their employer.
Happily, the Department of Labor exists and merely mentioning their name is frequently enough to scare an employer into compliance. I think a similar arrangement could probably be found to even the playing field in copyright, but finding the political willpower to get this done will probably prove difficult.
