Seems obvious when you say it like that,
That's exactly what happens. CSAM has two parts to it.
The first is the known and most horrific videos of CP out there. These aren't shared, instead only hashes are used to determine if the image/video matches. It's next to impossible to get an mis-hit on that.
The second is nudity and age detection AI. Apple has this on their devices, but warns the user before sharing not reporting.
It normally takes more than one hit to get flagged for investigation. At that point law enforcement are informed and take over.
The majority of investigations are children sending naked pictures of themself to their friends.
So it is absolutely being used as the purpose to protect children and it's amazing how many people don't realise how it works when they are very transparent about it.
Of course it is not without its issues that need to fixed. For example providers will just ban the person before it's determined if they broke a law or not.