It's unclear to me why it would be expensive. The hosting provider just has to check that a take down request includes all the statutorily required elements.

That should be pretty easy--they don't have to check that the claims are right. They just have to check that the claimant identified the infringed works, the infringing activity, and where it took place; check that contact information was provided; check that the claimant includes a statement they they have a good faith belief that this is infringement; and check that they have a statement that the information is accurate and they are authorized to make claims on behalf of the copyright owner.

Similar when responding to a counterclaim.

This seems like something that could be cheaply outsourced to some cheap forest digital sweatshop to handle.