No, running the executable after extracting the archive is fine, that's why I was confused about the Unity thing, it's not necessary just to run the game. If you're just getting a black screen on startup there might be an error of some kind but I'm not sure what would be happening that early, especially on a fresh download.
Can you upload the output_log.txt file contained in the game's save folder somewhere so I can check it for any errors? On Windows it's in the folder C:\Users\(your username)\AppData\LocalLow\Lazy Dog Software\LCS. You might need to enable viewing hidden files to see the AppData folder