In this case, the phone Support person was able to help me figure out the issue, but it's so tiny-weird that I figured I'd come document it here for the next person to run into it.
I wanted to schedule a refresh of my sandbox environment yesterday, expecting it to run overnight. However, when I brought up the calendar menu and selected the date 7/25/2023 as my refresh date, the 'Schedule Refresh' button became inoperable. I could click on it, but nothing happened and no error messages occurred.
I called in for help and we figured out that if you select the current date, Workfront assumes you want midnight of that exact date - ie the time that has already passed. Since Workfront has yet to install the Tardis time-travel feature, the UI silently stops you from scheduling the refresh for an 'invalid' date.
So if I wanted a refresh to run overnight tonight (7/26/23), then I would have to select 7/27/23 from the calendar picker to get my expected behavior.
Why the UI allows you to pick a date that will never be valid, or doesn't at least pop-up a helpful error, is a mystery for another day.