Thankfully said EU law includes a part that it's forbidden to block content for copyright reasons if the copyright claim is invalid. It includes ways for NGOs and users to go after companies that overblock. How this will actually work in practice is unclear since it's obviously an impossible requirement but some NGOs like the German GFF are already collecting cases and are looking to take legal action (see e.g. https://freiheitsrechte.org/aufruf-illegale-sperrungen/ (in German))