Got this game as part of the bundle for Ukraine. Ran into the exact same issue, I get "success" but no level unlocks.
The game seems fund but is completely broken at the moment.
So sorry about this, its an old project and maintaining it is a bit of a hassle. We'll get the standalone fixed again, but in the meantime you should be able to progress if you launch it through the itch app (https://itch.io/app)
Thanks for buying the bundle for Ukraine and thanks for letting us know about the issue!