Actually, the Post Office has some very serious legal mandates about not being allowed to open and inspect mail. You're going to have to get court orders, and I'd expect a lot of courts would ask "If you know there's illegal stuff being mailed, wouldn't it be more efficient to cut it at the source rather than try to intercept it once it enters the mail stream?"
Demanding a general block-- "you can't deliver any mail from Bob Smith"-- would be overbroad and silly-- he might be shipping meth, but he also mailed his electric bill and Christmas cards, none of which contain meth.
Here, we have a very similar issue with DNS. The DNS provider can't meaningfully know the intent of a given query; the site in question could contain both pirated content and cat videos, and there is no way to know which is being requested.
I also can't imagine how you'd expect this to scale-- if one firm realizes they can demand one domain removed, it creates precedent where eventually every cloud service provider and ISP is buried under requests. Even assuming every one of those requests is 100% legitimate, good faith, and accurate, it's simply going to be an untenable task. Inevitably, it would go back to the courts because the finite resources of service providers can't keep up with the tsunami and something got through.
The only possible way to make the Copyright Brigade happy would be to switch to an allowlist model: only these domains explicitly blessed by the Almighty Sony are allowed to be routed.
