You could do this now, without having to change any apps by using a library that listens to a folder and just commits any changes that happen in that folder after 2 seconds.
https://github.com/nevik/gitwatch
You could probaby modify it to auto unzip the zip file as well so that you'd have no manual steps after export.
In addition, I believe they are working on having export plugins, so in that case, all the logic for the extraction and git push could happen in an export plugin.