First, a platform with millions of posts from millions of users could not be further from small Web with its self-published, crafted websites.
Second, this law is precisely what makes it possible to run a social platform and not have an army of lawyers.
Without DMCA safe harbor protection, Nitter could be sued to oblivion the first time they are caught distributing infringing material. Big corporations with armies of lawyers and moderates could maybe afford the legal costs, but if you are just a few guys… you’d never run a website where people can post freely.
Under safe harbor, however, copyright owners can’t sue you, and in return you promise to timely hide content when you are notified. If you can’t be bothered to even do that, perhaps you should not run a platform focused on UGC in the first place. Everyone does it, even 4chan.