That is not what I have been told by multiple Concierge CMs. They pull from the same reservation pool you and I do. When a place is booked solid, it is booked solid.Isn't it not true that they might be able to get you an ADR at a restaurant in the resort you are staying at, assuming if it was sold out by checking yourself through the My Disney Experience app. A Club Level Cast Member once told me when they log into the reservation system it knows that it's doing it from a specific location like the Club Level lounge. So that's how they can find reservations even if a restaurant is showing as sold out to everyone else.
My own experience has show this to be fairly accurate. I have never had a time when a Concierge CM was able to get me an ADR that I could not get myself, even at a resort I was staying at.
What most of them do know how to do is work the reservation system to get it to spit out availability someone unfamiliar with the system might no be able to. (eg tweaking party size and booking multiple reservations, altering times, etc.)
There is also the possibility that a CM in Concierge knows a CM at the restaurant and can make a call to see if a walk up can be suck in if a restaurant is booked to the staffing level, but not necessary to the actual restaurant occupancy, but that is a case by case basis.