The other person who replied to you is wrong (edit: assuming they meant in general, not for only software that you have specifically already downloaded and run on a single computer before upgrading it to Catalina.)
Non-"notarized" software, when freshly downloaded to a computer running Catalina, will not run. A dialog will be displayed, telling the user that the developer needs to update their app for compatibility. There is no further indication telling the user what to do, and no "run anyway" button.
The developer will need to "notarize" the software with Apple, and receive their approval. If that happens, then the "notarization" information for the app bundle will be available via Apple's servers when users attempt to run the program, and Gatekeeper will attempt to look it up from the Apple's servers if the "notarization" information is not "stapled" into the app bundle. Optionally, and probably preferably, the app developer can "staple" the notarization information directly into the app bundle, and Gatekeeper won't need to look anything up over the internet the first time the user attempts to run the program.
They aren't "wrong" - sure there's no further indication telling the user what to do, but you can right-click and run the app from Finder all the same (requires admin permissions).
It's basically an upgraded Gatekeeper. I'm not quite sure what the pearl-clutching is about.
It's a problem for commercial software for end-users that isn't from the Mac App Store. Apple continues to tighten the leash, and it's approaching strangulation.
Please. I don't think you are replying to me in good faith. (Edit: on further reflection, I don't think I have anything useful to exchange with you. If you can't understand why this is a problem, there is nothing more we have to say to one another. Here's a hint, though: there is no regular menu item to do this. It's not discoverable at all, intentionally. Macs don't have a right-click button, and right-clicking or control clicking is not expected to be necessary to accomplish anything in macOS.)
I am...rather confused as to how you think people bring up context menus on macOS.
Unless you're being extremely pedantic and mean that I should call it "secondary-click" instead of "right-click", which wouldn't exactly be in the best of faith.
Apps that aren't distributed on the App Store are treated like second-class citizens on macOS. You can't even install them without having to chase down esoteric instructions.
If you don't understand why that would be an issue for a company that sells applications outside of the App Store, I don't know what to tell you.
No it won't stop working.
The only software that will stop working (and in 10.14 too) is software that was only signed and not notarized after 1 June. If the software was signed before 1 June, it will continue to work.
