As I have several sites, and Chat was working on some and not working on others, I investigated (same theme, same settings, installed other plugins one by one in a test site, I followed your flowchart), and discovered that, on one site, your Chat plugin doesn't appear on a page when this page displays the Facebook button from the Jetpack's Sharing plugin. It's funny because it works, even with that button, on a 3rd site but with an other list of activated plugins, so there must be a 3-ties interaction between some plugins. (EDIT: and it's the same with WP ajax or the plugin ajax)
I do not want you to tell me that I can use another plugin for sharing, I already know that :wink: and will either use another sharing plugin in the meantime or remove sharing from pages (I did that for today).
As being a member here is like "to have your own dev", and as Jetpack is used by millions, I think that if you solve the conflict, it will be great for next releases and your users.
Looking forward to test further with you.
I thank you very much in advance to have a look, and wish you a great day.