Well first to understand is that EU users will be effective by this not US. The US has other laws that clash with article 13 but still yes i can see how this is confusing. If EU law is pass then itch.io can do two things:
1) Make a US version and EU version
US version stay the same while EU version is more with the laws of that land.
2) Still remind worldwide with a requirement
So we developers have been under a honor system to credit people in our game. If the law pass itch.io can add like a credit system like in newgrounds to offset the law. There is no way for a robot to look into a game credit so it would be the responsibility of the developer to credit or else no game will get lunch.
The report button is still a good system because copyright holders can ask itch.io to remove it. That is the system EU wants. If a credit system like newgrounds is added it is easier for copyright owners to spot people who stole it. This can also be used to track games that use copyright owner's name, itch.io can have a system where the name of the person request to not be on any game he did not upload or something then a bot can take it down faster.
The only problems it would face is people uploading copyright images/video/music under fake names. Still that is something all sites face even youtube.