"Your system has extensions or settings installed that are incompatible with Private Relay"

Sorry to say but the possible damages CMM has created in the User Account and heaven knows where else, suggest restarting in Recovery Mode and choose to Reinstall macOS.

This will do an in-place installation, putting back elements of the OS that CMM has either corrupted or removed.

It should not touch your Personal Data, setting, preference etc.

To be safer - do a TM Backup before attempting or maybe 2 TM Backup each to separate external drives

Excellent catch by @Mal-S above.

Really do suggest following the Advise and remove " BrickMyMac " aka CleanMyMac as per the Developers Instructions.

Even at that, the damages this application may have inflected on this computer may required a Re-Installation to replace any corrupted or removed elements of the OS

Any of the below should be removed as per Developers Instructions

This will include CleanMyMac  and arrive at your own conclusions

Looking at the Apple support article doesn’t help other than to say that Private Relay is not available in all Counties and regions.

I do now suggest uninstalling the VPN as even though it may be turned off there may be associated helpers/daemons still in memory.

About iCloud Private Relay – Apple Support (UK)

Download the Application Etrecheck directly from the Developer. 

The application is free or paid from added features. 

It will take a Snap Shot -  both the hardware and software.

 The Report will Not Reveal Any Personal Information. 

Post back the Full Report - copy and paste - using the Additional Text Icon ( 3rd Icon to last )

We can have a look at the report for possible issues and may have possible suggestions to resolve the issues

2022-07-25 GlobalProtect (0)

2022-07-25 personalVPN (3.0.0)

Using com.paloaltonetworks.gp.pangpa.plist (Palo Alto Networks - installed 2022-07-25)

[Running] com.paloaltonetworks.gp.pangps.plist (Palo Alto Networks - installed 2022-07-25)

You are welcome and thank you too 🇨🇦

Do come back to the Apple Support Communities ( ASC )  in the future, if / or when additional questions may arise

Please post that report here, using the "additional text" button under your reply:

We may be able to spot something from it, as it often happens.

I’ll kick in after a quick scan of the report.

Uninstall CleanMyMac, it does more harm than good

Turn off VPN, it’s a competitor for Private Relay

Disable DuckDuckGo extension in Safari.

Offline and uninstalled are not synonymous. Whatever you are using to create a VPN connection may still be interfering.

Anything else set up like a proxy or other so-called internet “security?”

+1