I thought that was exactly how copyright works. It is infeasible to keep track of every infringement, so a policy like that is unenforceable.

That’s why DMCA takedowns exist, even automated after an Ai recognizes a song in example. As this has some bad side effects in an automated process or even persons and companies misusing it either willingly or by not understanding, it will remain a problem. DMCA exists because it is unfeasable to keep track of every infringement, and an instant action is taken just in case it is true.