Thanks for all your input. I'm still stuck, but here are my responses to the suggestions received so far (inline).
Crafty: Can you try requesting the ArcGIS Server Manager URL from a different machine rather than from the server directly? In other words, replace 'localhost' with your server name and then request it from your PC.
(I actually already tried that as noted in the op, but I just tried it again, with no luck. I�??m pretty sure the problem I�??m getting is from the arcgis server process, as it has that ESRI look to the �??Please Wait�?� dialog.)
bubbahey25: When I had that error, it was resolved by switching browsers. But you already did that. Also make sure you clear the cache and cookies for whatever browser you are using.
(Good idea, but it didn�??t change anything for me)
John Miller: Try stopping the service, then delete the server directories (contents of C:\arcgisserver), then restart the service to recreate them. Those directories may not have been fully removed when you uninstalled the first time, and could be causing a problem. It may also be worth checking to make sure the server user has full permissions on those directories after the install.
(I do not have these directories after the install by default (this is a fresh install on a new server), and restarting the service did not create them, but I created them manually and made sure the sever account has permissions. )
Michael Volz: Have you tried this installation on a Win2008 Server? I ask because if it works on a Win2008 Server, maybe it is a bug specific to a Win2012 Server.
(I don�??t have access to 2008 or r2 server at the moment�?� although I could try installing it on a desktop)
David Coley: I think you have share or port issues. Are you using a local or domain account for arcgisserver? Have you re-run the post-installation for the arcgisserver account? Have you checked all of your share permissions for the arcgisserer account for all of the system directories? Does that account have write access? Does your IT have any proxy or reverse proxy settings that you are not aware of? Are you running WebAdaptor on the same server? Is it listening on the correct ports?
(I am using a domain account, although I�??ve tried both. I have tried re-running the post-install. I�??ve checked all the account access I can think of, including giving it admin rights to rule that out. I am IT, so I�??m pretty sure IT hasn't set any settings I�??m not aware of. I am not running webadaptor. I am pretty sure it�??s not a port problem, because I am definitely getting an ArcServer spinning bar with the �??Please Wait�?� message, so I�??m hitting the web service.)
Inteller: I think we may be having the same problem. GO into your logs under the arcgisserver folder. Are you seeing entries that say something about "Unable to connect to the identity store using SSL." I ran post install, does nothing. I restored the arcgisserver folder to a week earlier, nothing.
(I have nothing in my arcgisserver logs folder�?� it�??s an empty directory. I do have logs at C:\Program Files\ArcGIS\Server\usr\logs but there are no errors in them.)