After an hour of troubleshooting, the app suddenly began working and showing the keyboard. We'll never know for sure, but the fix may have come by manipulating the Authorization method settings in the HOST PC's software, which acted to clear out an unseen corruption among the various authorization steps communicated between the Host PC and iPad app. Something had glitched and caused the iPad's app code to lock up just before it was supposed to pop-up the keyboard.
Here is the final series of steps taken that ended with the app beginning to work properly:
Remote Utilities Host software - Configuration menu: General, Dashboard, Authorization method section:
Experimenting with the Authorization method settings, I disabled the Single Password option, configured the "Remote Utilities security" option, then attempted and failed to login from the iPad. The next change was to disable the RU security option and try the "Windows security" option -which again failed to allow the iPad to login.
The iPad App's response to the RU security and Windows security options was a login error message that said something about that the proper login method was not selected in the HOST configuration, which was a very unclear way of saying that the App can only use the Single Password method for logging in.
Also to note: before each one of these password tests, I manually forced the iPad to unload its RU App from memory, so that it would have a fresh start at communications.
Finally, I re-selected the Single Password authorization option in the Host software and created a new password (forcing it to forget the previous password settings).
And then magically, the iPad's keyboard screen popped up! and allowed me to enter the password. It completed the connection to the remote PC and was in remote control mode.