What's producing the first animation? Not the Mac mini. Is it the monitor itself when searching for a connection to the Mac mini?
Also, try setting the monitor resolution, within macOS Catalina to something lower than your normal viewing preference. 1280x800 or there about and then switch it to the usual viewing preference of perhaps 1920x1080 or what ever you normally view everything at.
If you had anything dealing with resolution when using macOS Mojave then I suggest going through what you might have in your LaunchAgents and LaunchDaemons folders. Perhaps there's even a kext file that doesn't agree that well with macOS Catalina and can be updated somehow?
I've seen the jump in resolution on my Retina display iMac and the issues go away after I clear out NVRAM if SwitchResX has been installed - often I don't need to reset anything and just fiddle about in the Display Preferences.
The grey screen before login is probably something not appreciated with macOS Catalina