The way the “scan for existing game” feature works isn’t documented very well, I’ve put together a new page in the itch docs explaining what it does exactly: https://itch.io/docs/itch/using/scan-install-locations.html
Unfortunately at this time there’s no trivial way to convert a browser downloaded game into one the app understands, since there’s metadata about the game’s install that the app needs to track that is only created when the app installs a game.