Possible, but the Hitchhiker’s Guide makes a good point. If time travel was available at some point in the future. It would be no time until McDonalds existed in the Pleistocene.
“I went to a restaurant that serves ‘breakfast at any time’. So I ordered French Toast during the Renaissance.” - Steven Wright
And right across the street, a CVS, and a Walgreen’s.
Which suggests that Hitchhikers may actually be a documentation of historical events.
We know that McDonalds changes it menu from one country to another. Would they not also change menus to meet customer demands in any given era?
We know that there were around 150 thermopolium in Pompeii before it was destroyed. Burgers were prominent on those menus and some of the structures had prominent arches.
This would mean that McDonalds is working its way back to the Pleistocene but in our time line is still bogged down in the franchise wars of the Roman Empire.