https://www.apple.com/child-safety/pdf/CSAM_Detection_Techni...
Apple uses sophisticated cryptography to make absolutely certain that you cannot hold them accountable for abuses of this system against you, NONE of which are prevented by its complex construction.
The private set intersection is an alternative to sending you a list of bad-image hashes which uses significantly more bandwidth than simply sending you the list.
This alternative is superior for Apple because if they distributed the hashes it would be possible for someone to prove that they had begun matching against innocent images (such as ones connected to targeted races or religions, or sought to out particular pseudonyms by targeted images connected to them). It is inferior for the user for precisely the same reasons.
Some might be fooled into thinking the "threshold" behavior, somehow is in their interest: But no, Apple (or parties that have compromised them) can simply register the same images multiple times and bypass it and the privacy (for apple, but not for you) makes it impossible to detect that they've done that.