OneNote for Windows 10 won't work - perhaps because of lingering "ghost" installation

For a long time and across many reinstallations of my machine, I've had getting the OneNote windows 10 app to work.

Right from the beginning on a fresh Win 10 installation, the app doesn't do anything when run. Windows Store keeps reporting that there's an update for the app. But it never succeeds in installing it. It just fails with the following error:

The error code 0x80070705 doesn't seem to return anything useful if Google'd.

The fun thing is, that in the Start menu, the app looks to be installed just fine:

However, if I try to open the App Settings for the app I'm just sent to the general "Apps & Features" list. On the list of installed apps, it is also nowhere to be found.

If I list installed apps via PowerShell's Get-AppxPackage it's not found.

What to do, to get the app working again? I've filed numerous reports on this error over the past year.