That error appears to indicate a TSL/SSL error which could represent a certificate issue, a client-side issue with the http configuration of your app, or even a date/time issue (based on Stack Overflow questions and Gitbub.com/Xamarin issues).
It ended up being what I would consider an account issue, not an app issue, etc. We've been transitioning from the Esri Startup program to the ArcGIS Developer program and the Startup account settings were changed a month ago. Looking back, that's about the time that I started to see that Esri exception thrown on the Map.LoadAsync call.
Yesterday, I created a new Esri API Key as a test. Wouldn't you know it, the "Chain Link failed" exception went away and my call to Map.LoadAsync is now working again.