Unlike their competitors, Apple does not do Automatic Content Recognition (ACR). That matters to me.
Apple Intelligence takes a single screenshot of your screen when you explicitly direct it to. It does not take routine or scheduled screenshots and it does not exfiltrate or otherwise use the information other than to answer your query.
Private Cloud Compute goes to great lengths to prove it can't violate your privacy. In general, Apple likes to back itself up like this, which is one of the reasons I have such high confidence in them.
For example, on Mac, Command+Shift+5 will send the selected region to Siri once you enter a prompt to go with it. On iOS, after taking a screenshot, you can share it with Siri by tapping the "Ask" button. Neither of these are automatic or required, they're just alternative ways to use Siri.
Also none of these are sending the screenshot "to Apple" as you describe. PCC runs partially on Apple infrastructure, but provably (not a typo of "probably") does not allow Apple to see or review the actual queries or their responses.
Im pretty sure we managed just fine without SIRIs insights not that long ago.