Changing the hostname does effect the Site Import/Export job in the Scheduler. In the task information, the server's hostname is listed in the Servers box.  If you change the server hostname, then the Site Import/Export task will run, but the target will be the old server address. And the IMport/Export jobs will sit as scheduled but never run.

Updating the name there will allow the tasks to move from submitted to in progress, one at a time, when the Site Import/Export Scheduled task runs. You may need to kick it off manually to test it, and if you have a backlog of import/exports, may have to kick it off a few times (I did these exports consecutively, waiting until they completed before running the next export).