OK. I figured it out, Sebastian. I was using a skin that was suppressing the ViewState with the following property:
Within the following line:
<%@ Control language="vb" CodeBehind="~/admin/Skins/skin.vb" AutoEventWireup="false" EnableViewState ="False" Explicit="True" Inherits="DotNetNuke.UI.Skins.Skin" %>
Suppressing the ViewState shut down the ability to change Site Settings. It also disabled the ability to change Host Settings. The Host Settings issue was very funny, though. With no ViewState, when you try to update anything within Host Settings, you get this client-level error message:
"Address is invalid"
Really! It makes you think that your host email address is using an incorrect format even though it is. I saw a lot of messages from folks having an invalid email address issue within Host Settings, and a couple of them were unresolved. I wonder if these folks were having a problem with ViewState, as well.
After removing the "EnableViewState="False" property, everything works fine.
"But Van, why on earth would you have a skin that suppresses the ViewState?" Great question! It turns out that there's some legacy custom code on this site from 2007 that generates a ViewState-related error when more than page is loaded in different tabs. Suppressing the ViewState was a quick and dirty way of fixing the problem. I guess we'll actually have to dig in and rework the code.
Anyway, thanks your time!