Windows 10 task scheduler won't run any exe if set to "run whether user is logged in or not"

Solution 1:

What you describe is perfectly logical.

The "run a task whether the user is logged in or not" simply makes the task being run as SYSTEM user, rather than your own user to ensure that the task will run.

Given that you are not the SYSTEM user, the program will never appear to you, but it will still run.

Solution 2:

The applications that I was attempting to initiate via scheduled tasks were Office applications. It turns out that Microsoft has essentially switched off the ability to run Office applications using a System type account. They must run under a regular user profile. As far as I understand it now, this is the reason that the old tasks completely fail when the "run a task whether the user is logged in or not" switch is turned on. I didn't mention Microsoft Office in the original post because I didn't think it was relevant, but it appears to be the critical factor.