Okay, I've solved it -- and I can't believe I'm the first one to encounter the issue.
It turns out that, before the control would allow me to add states to the state tax field, I first had to populate the required fields (those fields that have form validation invoked, and that are marked with a red "*"), THEN SAVE THE CONFIGURATION FORM, then RETURN to add states to the state tax field. Obviously this was not duplicated on the test site because the cart has already been configured there and, presumably, all required fields previously populated. Make sense?
I respectfully suggest that is a bit of a design flaw, and one that unfortunately targets newbies setting up the cart for the first time (such as me).
IMHO, that control should either not be presented until the configuration is saved with the required fields, or the control itself should permit population of the data without having first to submit the form with the required fields populated. (Maybe it's an AJAX issue, dunno.) At the very least, this should be documented in the manual if it is not changed in a future build, IMHO.
Anyway, I hope this is useful feedback. :-)
Steve