A better way to make this claim would be to plot the mortality rates for the states over time and show that the rates in those places requiring reporting begin to diverge as the new policy is implemented.

They effectively did that: they compared public reporting states with other geographically adjacent states.

There's nothing like Boston or NYC in the control states. If the effect is as big as they seem to think--a 21% increase in mortality--that should be visible on a plot of deaths over time.