The way you push the app will not have any effect. If your game is launched through hhe itch.io app then it will generally avoid the Gatekeeper warning. If you want people to be able to run your game after downloading it directly off the internet without any warnings, though, you will need to notarize and sign your app file.
The advantage of using butler would be to upload only the changes of your game, and allow our app users to update their game via a patch instead of re-downloading the whole game.