Strenuously disagree. This is more than a billing error here, it is an implicit threat of expensive legal action wrapped in a takedown that at the least interferes with someone's free expression. They need to take it seriously or go away.
Civilization has long understood the dangers of crying wolf and even has a number of fables about it in order to teach young children not to do it.
They are welcome to use their algorithm as a screening test, but they shouldn't be claiming ownership of things without human verification. Since their algorithm must have some 'idea' what it thinks the work is, it should only take a few seconds per filing to have a human verify that what is playing is what the algorithm thinks it is.
Perhaps the ban on the easy method of making a claim should expire after a time, but the message is fairly clear: If they prove they are unable to responsibly use a largely automated system, they will be forced to do it manually in order to force them to consider each case more carefully. It may even be acceptable to grant them 3 strikes rather than 1, but only if they issue a personal (hand written) apology from their CEO to the person they wrongly claimed against.