I went to college near Cincinnati and developed my love of Cincinnati chili there. I'm surprised to hear Empress described as a "smaller chain" because it used to be second only to Skyline. Gold Star is the relative newcomer.
There is a good thread on the topic here.
Thirty years ago they didn't go too far from Cincinnati but today there are Skylines and Gold Stars all over Ohio, Kentucky, Indiana and perhaps other states. They're in Louisville, I know. Other places, in the Cincinnati area and elsewhere (even one here in Chicago) offer what they bill as "Cincinnati Chili," but from my experience only those three chains get it right.
The origin details are somewhat murky, as I think both Skyline and Empress, which are owned by different branches of the same family, claim to have been the first.
John and Linda's recipe for the dish is very good and authentic. I have made it many times. It is also very easy.
As for Covington trying to lay claim to some innovative variations, that is news to me, but I've never spent much time in Covington.
As mentioned above, beans are available as add-ons but the devotees rarely order them. The most common orders are the three-way or the four-way with onions. When you get it in the bowl (as opposed to a coney), you get oyster crackers. Hot sauce is usually available but also is not really part of the traditional experience.
It has always been understood that both the beans and the hot sauce are offered as a kind of concession to diners who want a more traditional Tex-Mex chili.
It's always been described as Greek and there are some similar meat sauces in Greek cuisine. I don't believe there is any Mexican connection.
Cincinnati, by the way, is a great town with a lot to recommend it. Not the least is how close it is to Kentucky.