First of all thanks to the detailed report!
I checked the status of your Jetpack connection, and your site seems to be properly connected to your WordPress.com account at the moment. I also checked the status of all 3 Publicize connections, and couldn't find any issues there either.
Do you publish all your posts by Email? And when you realize that it wasn't publicized, I assume that you re-publish them via the dashboard?
Could you try to publish a new post via the dashboard, and let me know if you still experience Publicize issues?