Fixed by following up on Conrad's diagnosis.
There was never any problem with Remote Utilities.
In summary, my modem was providing Wi-Fi in addition to the Wi-Fi from my router connected to the modem. I had my ISP bridge the modem and the direct access connection worked.
A few months ago I had RU direct connection working with no problem at all.
Between now and then I changed my cable, telephone, and Internet package. The old package did not include Wi-Fi; the new one did without my realizing it. And, I was too lazy to research this other network I noticed, and not smart enough to realize this other layer 1 segment could be the source of my direct connection problem.
But this software was well worth the time and effort to get it working. I have a strong preference for Remote Utilities over [censored] . I love this product and this company.
This is going to be an invaluable tool for a couple of commercial travelers who often need information (and sometimes functionality) from their unattended office machine. Additionally, it will save me much time and effort by allowing me to remotely support these fellows with their IT tasks instead of having to travel significant distance to their office.
Maybe a Google search will find this post and help others fix their direct connection problem or find a tremendous tool.