DNS doesn't work, though I can ping it. Adding a new one can solve the problem temporarily.

Are you running any anti-virus apps, cleaning apps, or third party security software that may be interfering with the normal operation of macOS? Try booting into Safe Mode which should disable Login Items and third party drivers.

Try creating a new user account. Log out of your main user account and log into the new user account. See how this new user account works first with the default macOS network settings, then after a few days try using your custom DNS settings.

Try reinstalling macOS over top of itself. If that fails, then perhaps you need to resort to a clean install by first erasing the drive before reinstalling macOS. Make sure to test your Mac before installing any third party apps and before migrating/restoring from a backup. Make sure to test the default macOS network settings, then just customize your DNS settings and test some more. It would be best to create new Network Locations for the various custom network settings so that you can change back and forth between the default settings and your custom settings.

I agree with @ku4hx's suggestion for configuring the router with the custom settings instead as that will automatically provide those same custom DNS settings to all of your other devices connected to the home network while keeping each device's network settings at system defaults which is a much safer & usually more stable setup.

My preferred method is to accept the Mac's defaults and change the DNS in my router. I've made that a practice for many years and have never had your problems.