As far as I am aware, there was a second road constructed between Armenia and the Nagorno-Karabakh Republic. That road is now within the territory of Azerbaijan, and nothing travels along it.
The November 9, 2020, statement by Russia, Armenia and Azerbaijan assumed that the connection between Armenia proper and the Nagorno-Karabakh Republic would be along this road. This was the route that the three parties to that announcement decided would be the connection that would be used.