The best way to increase privacy in the Apple ecosystem is to not use iCloud at all. Most of it (including your photos and backups, which contain endpoint keys and chat history) is effectively unencrypted and Apple can read all of it at any time without your device. Apple intentionally preserves this encryption backdoor in iMessage/Photos/iCloud for the US federal government, who can then access this information without a search warrant or probable cause.

This means creating a burner phone number, using that to create a burner Apple ID that is used only for installing apps, using only free apps (because the moment you put your payment card information in, you're deanonymized) and only using devices bought for cash.

Then Apple has a fair amount of information about you, but it's not linked to your identity.

Apple turns over customer data to the US federal authorities without a warrant over 30,000 times per year per their own transparency report. This is in addition to the normal legal process stuff that involves subpoenas or probable cause-based search warrants.