My latest outcome from troubleshooting this issue:
tested the last 12 hours and can confirm that BLOCK_ICLOUD_PR=false in pihole-FTL.conf is a valid workaround for now. Safari and Mail on iOS 15.5 and macOS 12.4 are now working as expected, all my affected websites are loading correctly.
My iOS/macOS settings are (which is Apples default setting):
Private Relay = OFF
Limit IP Address Tracking in WiFi settings = ON
Adblocking via PiHole is also working as expected on these Apple devices, I can see the blocked URLs in the PiHole log correctly.
Looks for me that the newest releases of iOS and macOS are pissed if anyone is blocking Apples DNS servers and they cannot be reached. 
FYI: I have tested the following FTL settings:
- NO BLOCK_ICLOUD_PR config in pihole-FTL.conf (PiHole default)
- SET BLOCK_ICLOUD_PR config in pihole-FTL.conf to true
- SET BLOCK_ICLOUD_PR config in pihole-FTL.conf to false
Only TEST3 with value false is working without issues in combination with Apples default settings on iOS / macOS.
Posted this also to: https://www.reddit.com/r/pihole/comments/utab8p/safari_wont_finish_loading_certain_sites_after/