Thanks to those who offered suggestions!
What happened to our Enterprise Deployment may be 'non-standard.' But I will explain as best as I can understand it myself:
When we were still at the version 10.8.1 state, we transferred the whole deployment from one "box" to another. This was part of the SQL Server upgrade, and our IT Section's management of data-space; it was done on the advice of an ESRI Technician and with their guidance. After upgrading the 10.6.1 Deployment to 10.8.1 on the original "box," we created a new 10.8.1 Deployment on the new "box," then joined the new site with the old site while we copied things over [* I'm positive that "copying things over" isn't what we did, but that's what it felt like]. ...Whether or not we left the old "box" running at this time, I'm unsure, but would guess we did (considering the problem we discovered today).
Then we upgraded to 11.1, and attempted to go upon our merry way. ☹️
What we think happened (posited by an extremely helpful ESRI Tech, Dana A.):
During the time the two Enterprises were joined, some internal confusion happened wherein our deployment confused which site was the primary and which the standby. While it is possible to unregister the standby machine, you can't unregister the primary. ...But when we were done with the joint operations and un-joined, the confused configuration attributes still recorded the old "box" as the primary (master node).
How was it discovered:
The arcgisportal\content\items\portal\portal-config.properties file was opened, and compared with the "sharedkey" value in Program Files\ArcGIS\Portal\framework\etc\portal-config.properties and Program Files\ArcGIS\Portal\framework\webapps\arcgis#sharing\WEB-INF\classes\resources\gw-config.properties.
[Honestly, there was a lot more to it, but since even I could tell that something was off here, I'm adding it to this explanation.]
The Fix:
Yeah.... it was a truly dizzying succession of tracking down configuration files from everywhere, opening them up in Notepad, and altering the master node and key files attributes. Our IT Admin and the ESRI Tech also queried the Log files and watched processes resolve in Task Manager. While using the template of a "healthy one machine Portal for ArcGIS site" as the goal.
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The ESRI Technician who helped us resolve this issue was incredibly professional, proficient, and patient. She is apparently part of a Technical Support Office in Jordan. If any of you are having trouble with your Enterprise Deployment, I'd recommend trying to get ahold of someone in that office.