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Flash used to work on the desktop as well, they had this little "projector applications" iirc

They still do, but if you want a Flash-based desktop app, your best bet is Adobe AIR, which gives it access to native APIs.

You could "compile" your animation as an executable file, as well as an SWF file.

you still can, it is called a "projector", you basically merge the Flash Standalone player exe with a SWF into one executable.