Did you ever get an answer to this question of how to publish a connection when the option is not there to "publish to server"? I need to be able to take an existing connection and republish with new name. The remove the old connection as we now have naming standards.
Tableau Support provided the following steps to support the creation of an aggregated TDE that can be re-published and set to refresh (and these steps do address the reported issue). I would strongly prefer an approach similar to the Extract Children idea.
This may a little different than the use case you are describing.
1) Connect to the published data source from Tableau Desktop
2) Create a local copy of the data source by selecting Data > <data source? > Create Local Copy
3) Save the local copy of the data source
4) Replace the published data source in the workbook with the newly created local data source by selecting Data > Replace Data Source
5) Close the published data source
6) Pull the aggregated extract from the Local copy
7) Publish the aggregated extract with the new name
8) Set a refresh schedule for the newly published aggregated extract.