MM hedge delta using stock (in most cases). So, if an MM sells a call, he will instantly buy stock to cover delta. Initial hedging has nothign to do with his convexity position, it's simply negating the directional risk.What is not a difficult concept is the fact that someone has to take the other side of the position for you to hedge. Who?
Now, if MMs were drastically short calls and long puts, they would flatten the skew to trade out of this imbalance. Looking at the vol surface, I do not see anything like that for yesterday or today.