If you exhaustively performed all the recommendations in that Apple Support document, I doubt a back up and restore will change anything. As inconvenient as it may be, I'd contact Apple again and ask them to help.
I've never seen a chat simply disappear; you ought to at least have had the opportunity to retain it before exiting the chat. I realize it doesn't help to know that, but something isn't right. Apple will be more than happy to help fix whatever's wrong, especially since your iPhone is new. There may still be something amiss with that website, which is something no one else can fix.
When contacting Apple, be sure to use the Support link that appears on these webpages. That's by far the best way, despite the fact you have to jump through some hoops to get the help you need.