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.
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.