(First off: Thanks for purchasing itch's Racial Justice bundle!)
Huh, that's a brand new one for me. As far as I've seen, the game has never crashed a whole computer before, even during development when there were a lot of crashes to go around! Have you had this issue with other games or apps?
Some technical background: The game doesn't "install" exactly. Like many games, the contents of a folder are simply copied to a directory where you run the game from. When you install it from the Itch desktop app, it works similarly, it just chooses the directory to copy to for you. By default, this is: C:\Users\[user]\AppData\Roaming\itch\apps\widget-satchel\
The game itself has no installer executable, and does not really interact with the operating system. It doesn't create new icons or files or anything like that the first time you run it. This is the same if you manually download the app as a ZIP, or copy it to your system using a client like Itch or Steam. So, the fact that the app icon itself disappears when you try to run it is very troubling, because I can't imagine that the game itself is doing that.
But let's try to solve it! Here are some possible troubleshooting avenues...
When you say "the icon that would be used to launch the game" what do you mean by that? The app package does not contain any shortcut icons. The icon you click is the game's executable, so if that disappears, it's your OS deleting a file! Yikes! Where exactly on your system is this icon?
Are you using a 64-bit or a 32-bit operating system? Widget Satchel is a 64-bit game on all platforms. The way you can check in Windows 10 (as well as Windows 8 or 7) is by hitting the Windows key on your keyboard and typing "About," which finds the About your PC panel/window. Under "System Type," you should see "64-bit operating system. x64-based processor." If you see "32-bit operating system," you won't be able to run Widget Satchel.
Even though 32-bit operating systems are rare these days, many games, from indie to AAA, still are 32-bit because they run fine on 64-bit machines. Some games will offer two versions, so you might not have noticed this if you don't have a lot of games on your machine. If there's enough interest, I can look into releasing a 32-bit version (no promises!).
It's possible that this is an issue with the graphics drivers or support on your machine. Widget Satchel should select the correct graphics API based on your computer's best compatibility. On most Windows systems, that's DirectX 11. It's possible that it's selecting a graphics API that it thinks your machine supports but doesn't. This is something that I've seen happen with other games on systems with Intel integrated graphics. My wife was just having this problem running Civilization VI on her Surface Pro. Long-shot, but maybe a few system updates or driver refrashes might help?
The game uses your "AppData" folder to save settings and game data. The directory is: C:\Users\[user]\AppData\LocalLow\Noble Robot\Widget Satchel\ (enable "show hidden folders" to browse to it, or simply copy the directory into the address bar in File Explorer). You can check that directory to see if anything made it in there, or you can try deleting the folder it before "re-insalling" the game, although there shouldn't be a reason to. It's possible that your system is preventing writes to this directory (or the hard drive is full) in which case the game might crash. However, if it were the case, you'd be having lot of other problems since tons of apps write to that directory.
-----
That's all I can think of to try right now. As I said, I've not seen anything like this happen previously, so I'm assuming it's something that you'd also see when running similar games. But of course we can't ever be 100% sure it isn't Sprocket's doing!
Let me know what you discover.