What should Tor Browser Updater (Whonix) do in a TemplateVM?

That will be the case in Whonix 12. Desktop shortcuts renamed to include ‘downloader’. Same goes for documentation, window title and window contents.

Here is an older screenshot (without the Qubes specific message):
https://phabricator.whonix.org/file/data/epd4vbnqym63tajhww7p/PHID-FILE-fluqfju3qgvt7hv5wbvx/tb_updater_local_version_numer_detection_not_implemented.png

Here is the text with the Qubes TemplateVM specific message:

Here is a little more background to all of that:
https://phabricator.whonix.org/T400

No.
Just now wrote a feature request…

headless TBB updates [by script]
https://trac.torproject.org/projects/tor/ticket/17136

Created a new ticket.

up to date versions of Tor Browsers in newly created AppVMs inherited from updated TemplateVMs
ship Tor Browser tarballs in Qubes TemplateVMs in /var/cache/tb-binary and extract in AppVMs at boot time to user's home folder
https://phabricator.whonix.org/T417

The above ticket up to date versions of Tor Browsers in newly created AppVMs inherited from updated TemplateVMs has been implemented and it’s now documented here:
Tor Browser Essentials