Technically, you could use the Tableau Data Extract API to create a TDE file, and then use the REST API to publish it to Tableau Server. Assuming the Data Extract API doesn't require a license to Tableau Desktop (check the agreement, I have no idea)
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.
Seems like a lot of effort to avoid buying a few Desktop licenses which are useful in their own right. Editing workbooks in Tableau Desktop is generally 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.

1. Tableau Desktop is a client-based tool with software you need to install.
2. Tableau Server and Tableau Online are browser-based tools (no software installed).
3. Tableau Server has most of the functionality required to create data sources, dashboards and workbooks, but is still missing the ability to create and modify dashboard actions.
4. The Cluster and Forecast options in the Analytics pane are not available in Tableau Server/Online. At this time, these options are only available in Tableau Desktop.
5. With Tableau Server, you have a Username which is assigned a Site Role and a set of permissions. The functionality available to you depends on the Site Role and the permissions granted (usually by the Tableau Server Administrator).
6. The mechanisms for saving and publishing are slightly different between Tableau Server/Online and Tableau Desktop, but the results are the same. With Tableau Desktop you can publish workbooks/dashboards/worksheets to Tableau Server/Online or you can save them as a packaged workbook (twbx). Packaged workbooks can be opened in Tableau Reader or Tableau Desktop. In Tableau Server you save your worksheets to projects and folders defined in Tableau Server or you can download and save them as a packaged workbook. (NOTE: Download and Save are permissions that are sometimes not granted to prevent unauthorized downloading of workbooks and data.)
7. Even though you can now create a new data source now in  Tableau Server and Tableau Online, the number of "connectors" is limited.  In 2018.2 of Tableau Online, there are only 10 different connectors whereas in Tableau Desktop there are over 70.  (Likely Tableau will be adding to the list of connectors with every release). 

Yes, to the best of my knowledge you should be able to publish without going through Tableau Desktop using the REST API. See the link below for more details. Provided you are assigned a role with publishing privileges for one or more projects on Tableau Server, whether or not you publish directly from Tableau Desktop shouldn't matter.