Since a complete wipe and fresh reinstallation of the operating system did not fix the problem, the issue is not caused by corrupted local files or standard app preferences. Instead, this behavior usually points to either a specialized system permission glitch, network-level security blocks, or a hidden proxy setting interfering with Apple’s identity handshake.
Try the following targeted fix to resolve the issue:
Disable Auto Proxy Discovery
A known bug in some macOS versions causes secure connection timeouts with Apple Identity Services if hidden proxy flags are active—even if you don't intentionally use a proxy. Turning them off can immediately stop the app from crashing:
- Open System Settings and go to Network.
- Click on your active network connection (e.g., Wi-Fi) and select Details....
- Click on the Proxies tab in the sidebar.
- Turn OFF all toggle switches, specifically Auto proxy discovery and Automatic proxy configuration.
- Click OK, restart your Mac, and test both apps

See if the problem is present when you boot in safe mode. That process performs some system cleanup and clears caches, and also disables 3rd party extensions that may cause problems.
Start up your Mac in safe mode - Apple Support
More info from Apple: If you can’t turn on or sign in to iMessage or FaceTime on iPad, Mac, or Apple Vision Pro - Apple Support