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from talking to apple support and reading through this thread, what worked for me was:

going into settings, general, login items, and turning off my vpn there. then rebooting into safe mode. in safe mode the apple id pane worked fine and i was able to turn off iCloud private relay. then rebooted like normal and turned back on the vpn in login items and all worked well.

so private relay is the issue, didn't need to install vpn. hopefully its a bug that gets fixed rather than a choice.
 
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I have been struggling with this also. Private relay and and my VPN (Express VPN) worked well together before updating to MacOS Ventura. Private relay was always active until I launched Express Vpn at which time I would get the notice that it was no longer active. Now I don't even have to launch Express VPN and it causes problems with iCloud settings hang ups, so for the time being I have uninstalled the VPN until an update from either Apple or Express VPN fixes the problem.
 
Problem started for me when I updated to Ventura 13.1. I had no VPN running and had uninstalled ExpressVPN long ago. But I searched for and deleted all remaining mention of ExpressVPN on my Mac and now can use Private Relay again. Its plist file was one of the holdouts.
 
I'm using Mullvad, the only workaround I've found is to disable Private Relay (not for one day, but to completely disable it), then reinstall the VPN app and everything works, System Settings doesn't hang, iCloud syncs, Apple Music works.

The Mullvad app is 2022.5, which is still the version that was creating problems.

Of course with the VPN running I lose access to my beloved AirDrop, but even with disconnecting and reconnecting the VPN, AirDrop has problems resuming working, but I think it's normal since my iPhone and my Mac would basically be on two different connections when I'm connected to the VPN.
 
I received an answer (!) from the feedback I sent through Feedback Assistant:

There are changes in the latest update, build 22D5038i, that may have resolved this issue. You can see the software build your device is running and check for the latest update by clicking on the Apple logo in the upper left hand corner > About This Mac. If the build is not visible, click on the macOS version, e.g. 10.15.x, to reveal it. Has this issue been resolved after installing the latest update? If not, please use Feedback Assistant to let us know you are still experiencing it.

In case anyone has installed the latest macOS beta...
 
This is happening with my M1 Macbook Pro too. The iCloud section in System settings just freezes while all other sections work perfectly fine. Restarted numerous times....doesnt help. Only after reading this forum I got to know the solution. I had to uninstall the 3rd Party VPN...then and only then the issue for fixed. Really weird as before this update of 13.1 everything was working fine. Hope apple releases the fix asap.
This issue is fixed now since 13.2
 
I have Private Relay turned off, and I still can't get any third-party VPN client to work (ExpressVPN or NordVPN). Pretty sure this issue only happened with 13.2 though.
 
I have Private Relay turned off, and I still can't get any third-party VPN client to work (ExpressVPN or NordVPN). Pretty sure this issue only happened with 13.2 though.
I'm running Mullvad VPN, it runs fine even with Private Relay turned on, using the official Wireguard app from the Appstore.
 
Just a heads up -- I am running 13.4 with no VPN and still getting this problem. I used bogdanw's suggestion of creating a profile to manually disable Private Relay, but Settings > iCloud is still broken for me (it's just blank).

I have tried safe mode, running repair assistant in recovery mode and even doing a fresh install of macOS. Issue persists...
 
For me, in the end, Apple Support had me just wipe my computer completely. Uninstalling stuff, fresh macOS install, etc did not solve the problem. But the moment I did Erase All Content and Settings and reinstalled exactly all the same apps and such, it fixed it.
 
I have Private Relay active and Torguard VPN connected at the same time. The only issue I experience is an occasional loss of internet connection. I simply disconnect Torguard and re-connect. I'm thinking of starting a thread for this issue. Anyone else experiences this?
 
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