you are correct that this endpoint does not exist. I see two solutions to this:
If you have already access to the enterprise SDK you can “simply” create your own extension that adds your own REST endpoint (this need the enterprise SDK) that then uses the PublishService.publish() (this one is in the public SDK). Note that access to the enterprise SDK needs a certification (which costs money).
You don’t want to spend money → Again you need to create your own extension that implements a REST server where you then basically create your own endpoint that again uses the PublishService.