SuperDuper won't "register" for notifications until it has notified. So you have to quit Growl to get us to notify. We definitely do notify with the notifications center as long as Growl isn't running: I've had no other reports of it not working, so I'm not sure why that would be...
I've tested Growl and haven't had any problem under Mojave or High Sierra. My guess as to why it's been removed from the App Store is that it's available for free outside it. You can go to
http://www.growl.info to download the most recent version. And you can contact them for support...