Default browser selection completely broken after Monterey 12.5 upgrade

After upgrading to macOS Monterey 12.5, default browser registration and selection is completely broken.

Symptoms:

• System Preferences > General only lists Safari, Brave Browser, and Wavebox.
• Registering a new default browser from within Firefox or Chrome brings up the correct "Use Chrome" or "Use Firefox" system dialog box, and Firefox even reports it is my default browser. But neither browser gets listed in System Preferences > General.
• Even if I do select Brave Browser as my default browser, opening a URL from any app (like Slack or Discord) opens the URL in Safari, not Brave.

In other words, my system is now locked to Safari as a default browser and I cannot find any way to change it.

Is there any way to force a browser to register with the OS via the command line? Or is my best bet to just wipe and reinstall Monterey clean?

I'm on a Core i9 16-inch MacBook Pro, 2019.