Am I doing something wrong? Why not restart the application in step 2 with the highest available privileges? In my mind, that is the behavior I'm supposed to get when I actively choses to Run the software as administrator in the first place.
You're not doing anything wrong, and this is by design. wyUpdate restarts your app as normal elevation. Why? Because very few apps should be running as elevated. (Also, "normal" apps experience problems when running elevated. For instance, drag & drop between explorer fails to work).
I want to be able to set the application in a "fire and forget" mode, where I start elevated, and let the application to auto update itself, forever. Is there a way to accomplish this (except placing the application in a "user profile folder"?.
Other than putting wyUpdate in the user profile folder? Yes. Either install your updates using a dummy Windows Service that calls wyUpdate (see: this or this). Or give non-elevated users permission to modify your files and wyUpdate won't need elevation.