New mail rules disappearing after closing Mail app

Download and run EtreCheck, and share the report to the clipboard. Then open a new reply here, and press the Additional Text button that looks like a printed page to get a text input box big enough to paste the hardware and software configuration report here, and paste and post it here.

Remove all of the add-on anti-malware apps, remove Clean My Mac, remove the VPN, remove AdGuard, and all removals per the vendors’ instructions, restart, and try again. All of these are of the sorts of apps that can cause as many or more issues as they might resolve. Cleaning apps can also sometimes cause latent corruptions.

Add-on first-few-hops VPNs are too often a privacy disaster, and too many badly solved a problem that hasn’t existed for a decade or so. (Too many of the add-on security apps are similarly problematic.) If you want some browsing privacy, the existing iCloud+ Private Relay feature might interest, as that uses the existing VPNs (the add-on VPNs add a second and weak VPN around the first and much stronger VPNs, and adds that only for the first few hops), and adds a VPN for DNS traffic.

I tried deleting and reinstalling the email accounts to no good end. rebooted. Set up additional user account and the problem did not exist - everything normal and rule remained. Went to Internet Accounts in Settings in my original user account and turned off every email account and rebooted. Returned to Mail, turned on all Internet accounts and entered new rule. Rebooted. checked settings again and the new rule remained. Rebooted and new rule was gone. I had to do this process 2-3 times to fix the issue (no explanation at all why repeating the process fixed the issue). So far it is remaining as an operative rule.

There has been the appearance of some very low level iCloud corruption in very occasional cases, both sometimes locally and in iCloud. I’ve seen maybe a half dozen cases over the past decade or so, across a whole bunch of systems.

The fix for those cache incoherency tends to be setting that feature to the other way or to the wrong way or whatever, restarting and quite possibly restarting Macs via Safe Mode, and re-enabling the feature, across all systems sharing the Apple ID. This usually drags things back into synchronization.

And if there are add-on security apps including the heavily-hyped first-few-hops VPNs, those can tangle or block networking, and can get things further tangled.

Updating to current macOS is always in play too, as Apple tends to fix bugs quietly.