AFAIK, the DMCA doesn't require infrastructure providers (ISPs, DNS resolvers, "relay" services like Cloudflare) to block entire websites. It's just for surgical removals of content (and blocking of ISP / hosting provider customers who are notorious infringers).

The US doesn't have the kind of website blocking laws that many European countries have.

If you look at those 'whole websites' it is nearly exclusively sites that do not comply with takedown requests regarding copyright (actually those blocking laws/procedures do mostly foresee any other reason). The question I was addressing is the judicial control and the abuse for censorship. DCMA takedown request are massively abused without any real judicial control. Sure you can fight those in court, but so you could fight ISP blocks. I thing the different methods simply stem from a different legal system with different types of fines (particularly in civil law)