CLOSING: I'm closing this question based on discussion in the comments and starting over from a fresh perspective.
On a Windows 2008 server with IIS-hosted websites, I've never been able to access those sites from within that machine. I never wound up solving the problem. Now I'm trying to set up Windows 2016 for the same websites, and I'm finding the same problem, using Internet Explorer 11. I can access those sites from other machines on my network, but not within the server hosting them.
What could be causing this?