QV simply is not allowing some actions / macros to be executed on the server. So there was nothing wrong with my settings, it simply does not work by design. It would be helpful for QV to display some kind of message, or put something in the log for this case, but these guys in Lund are not very concerned about this kind of issues, IMHO.
So the solution in my case was to use EDX load. It requires additional coding and task's settings update, but it's the only officially approved way to do this.
There is a ton of information on the forum and in documentation about this method (for example: Using EDX in QlikView 11), you should be able to find few articles about the process.
The Reload action is not allowed when running in the server environment. If you want a sheet button, you'll have to use something that uses the EDX facility to trigger a task on the server. Here's one approach.