They don’t. Everyone else calls them “Modern” so the name stuck. A real, knowledgeable “MO” person would most likely refer to themselves as “Centrist Orthodox” or (ironically based on the path of this thread) “Authentic Orthodox”.
It’s the same as Chalav Yisrael. People spent their entire lives trying to convince me that I think Chalav Yisrael doesn’t apply anymore. Chas V’shalom. My Rabbis were very particular in saying that of course Chalav Yisrael still applies. It’s just that in America because of FDA regulations it’s like all Chalav Stam is really Chalav Yisrael. No Jew, MO or otherwise, would deny an Issur D’Rabannan. They just say that the current level of supervision fulfills the requirements of that Halacha.