You already know the answer.
InDesign Server is not the product you have - and Server costs far, far more than your desktop license.
There are a few different ways to approach your project, but the one you want ain't it.
InDesign can run startup scripts that you can build with Extendscript (comes with InDesign.) It might be possible to dedicate an InDesign installation to start up with a dedicated script that performs what you need, when your source-control application launches InDesign.
Another possibility might be to have an InDesign script periodically look for a file in a location and when it finds it, perform operations on that file. The source-control application would put the file for InDesign to find.
These are technically called "wild guesses" at solutions. You sound tech-enough to go further.
However, read the InDesign license agreement to see if this is permissible use. Some applications don't permit using a single-user installation as a service for other users.
[EDIT] I forgot to mention that InDesign does have some XML smarts. Depending on your needs, it might be possible to output InDesign documents and/or templates in XML form, and process them with an XML processor that does run from a CLI. Worth a look, anyway. Also, look up the book XML for InDesign by James Maivald. You might find useful info in it.[/EDIT]
Message was edited by: peter at knowhowpro
Thank you everyone for your valued input.
After experimenting a bit with automating InDesign to work with git and shell scripts... it proved too complex to waste any more time on.