Same issue here, it doesn't seem to play well with the itch app.
I guess that's because itch app expects a standalone build for your OS, which is paid that's why you don't have access. I'll try to find a solution with the help of itch support, but I need more details from you. You didn't purchase a full version of the game, right? The game is downloaded, but you can't run it because of the 'no permission' error? If you could provide a screenshot as well, that would be great.
I was able to run the game on my Mac with this trick given by the OSX user guide (which also means you trust Andriy 100% not running any malicious software with the game):
Open an app by overriding security settings
You can open an app that isn’t allowed to open by manually overriding the settings in Security & Privacy preferences.
- In the Finder
on your Mac, locate the app you want to open. Most apps can be found in the Applications folder.
- Control-click the app icon, then choose Open from the shortcut menu.
- Click Open.
The app is saved as an exception to your security settings, and you can open it in the future by double-clicking it, just as you can any authorised app.