Good news:
The tech and I identified the issue and workaround for this in our case. In some cases, the NSPackageRefresh scheduled task hangs and never completes. When this occurs, you start seeing the “Unable to Map HTTP Virtual Directory” errors in the NS logs and clients then get a blank package source at the client and consequently cannot download packages. The package.xml in the Software Delivery folder on the client will say "
statusDescription="No package sources returned by server".
As a workaround, you can do a full server reboot. Although not confirmed, you may be able to stop the task, then restart IIS to get things functioning again. You need to be mindful of the package refresh task though and verify it is not hanging after its next run.
I assume this is being treated as a bug at this point by engineering. I'm just happy I have a workaround.