As far as I can tell you can use tableau server to make worksheets based on other worksheets, but I can't figure out how to add a new data source and create an initial worksheet without using the desktop version at all.
You can use the Tableau Data Extract API to create a TDE file, and then use the REST API to publish it to Tableau Server. The Data Extract API need not require a license to Tableau Desktop
You could probably do the same with a TDS file, which is just XML, so you could presumably create that by hand or by script if you figured out the format. But no guarantees that the format won't change with future versions.
It takes a lot of effort to avoid buying a few Desktop licenses which are useful in their own right. Generally, editing workbooks in Tableau Desktop is a faster more fluid experience than via the web, so having a few Desktop licenses is usually well worth the cost.
But if your goal is to automate a manual process, that's exactly what the Data Extract and REST APIs are for.