Strange thing number 2 is that even the example from Construct editor.construct.net is not working. It can unlock levels, but the clear button does not work until the project is reopened. I even tried to force restart the layout by pressing the button after clearing, but nothing changed.
The layout is restarted already, but the array is not cleared. Simply add this action to the last event:
-> System: Set gameFirstExec to True
This will force reading the array from LocalStorage, and because the key is removed from LocalStorage, the array will be filled with default values.