These days you can pull Facebook Messenger notifications into other platforms, including Intercom. It also works with FrontApp and Sprout Social I believe. So they wouldn’t have to be on Facebook to monitor, but they would have to be signed up for one of those other tools, which might not be ideal.
In my experience, the problem with embedding a chat window on a page is that visitors like to navigate to other pages, and in those cases, they would lose the chat. So that’s why the little popup chat icons in the lower corners of sites are so popular.
As a middle ground approach, one idea is to just have call to action buttons on your page that when clicked, launch the popup chat. I know you can do this with Intercom, and probably many other common chat platforms.