Nor are there any "After applying package" tasks that I know of, other than standard Applicability and Installed Rules. Where should I look so I can verify?
YOU created one! To perform the XCOPY of the XML file! :-)
I added the "Continue with next action" to the "Install anyconnect" step.
What should I do to ensure that the "After applying package" tasks complete?
Publish the package. Ensure the client has detected the new revision. Install the package. Observe the results.
By setting the option to "Continue with next action", the "After applying package" tasks will ALWAYS execute.
Then, knowing for a fact that they ARE executing, we can further diagnose whether its executing correctly, or not.