And "Big Copyright" prefers it that way. They are only interested in maximizing sensitivity (probability of taking down a true violation), they have no incentive to care about the trade-off with precision (probability that a taken-down violation is a true violation).

They process incorrectly filed notices too, for example notices that don't contain the information required. I had my content taken down once in response to a notice that just had a person's name and no other information (they were not the rightsholder)

The service provider is only liable if an infraction has actually taken place. Checking DMCA claims means determining if that is the case. Service providers do have that leeway.