Problem solved:
The problem I encountered with SteerMouse is the terminology used its control panel interface. When you set "Primary Click", you'd expect this to mean a normal, simple click of a mouse button, but no. This apparently only happens with "Secondary Click".
When I gave then left and right mouse buttons the "Secondary Click" option, then they finally started functioning properly.
However, one of your mouse buttons must be given the "Primary Click" option -- SteerMouse insists on this because it assumes you will not have a functioning mouse if none of your buttons are set to "Primary Click". I chose to give the "Primary Click" option
to the scroll wheel as I have never used the click option on the scroll wheel before.
So long story short, giving "Secondary Click" options to both left and right mouse buttons gave me back my mouse.
This is not entirely true because my mouse no longer can change tool features in
Abobe Illustrator with one click. I now have to toggle the Option Key and Mouse and the click-scroll through each tool to get the tool I am looking for. This feels like an odd hoop to have to jump through, and a tad bit annoying, but I think I can
start doing it quickly and easily with a bit of practice. At least I have my IntelliMouse back, that is a big deal... at least until OSX10.12 comes out.