Far more likely Apple takes a bunch of hashes from a third party in the law enforcement side of things (ie cops) and trust that the third party is definitely giving them hashes to protect against the Very Bad Thing that Apple's customers are worried about.
Whereupon what you're actually trusting isn't Tim Cook, it's a cop. I'm told there are good cops. Maybe all this is done exclusively by good cops. For now.
Now, I don't know about the USA, but around here we don't let cops just snoop about in our stuff, on the off-chance that by doing so they might find kiddie porn. So it should be striking that apparently Apple expects you to be OK with that.
It _does_ make sense client side if you view it being done server side as a blocker for E2EE on iCloud. There is absolutely no world where Apple could implement that without keeping the ability to say "yes, we're blocking child porn".