Thanks to another reader (known only to me by 'First Last') the login issues have been resolved!
It turns out that the problem was much more simple than I originally thought. 'First Last' pointed out that the query string parameter 'service' needed to be passed along with the value of 'grandcentral' (which is the name of the company that Google acquired in order to bring us Google Voice).
The change has been made, and the latest versions of the various Google Voices script can be downloaded from my github repo.
From now on, my scripts will be available at that location. It is much easier for me to maintain, and for users of github is it much easier for you to keep up with the latest version and contribute back if you wish (which would be nice!).