Embed Facebook Customer Chat Plugin without creating a Facebook app

It seems in order to embed Facebook Customer Chat Plugin, dev has to create a custom Facebook app to act as hook receiver to be called from the embed. The flow is somehow this way:

Embed on browser ---custom app id,page id---> Custom web server ---page access token,text response--> Facebook Messenger Server --> Back to Embed on browser.

This is good for making a chatbot but can it be simpler to chat directly between visitor and a Facebook page owner, this way:?

Embed on browser ---Messenger app id,page id---> Facebook Messenger Server ---> Page owner on Messenger.com ---text response--> Embed on browser.

Facebook Message Us Plugin is another option but it opens a new tab. Overlay embed is more interesting as visitors are usually lazy to chat in a new tab.