In reading through that breif, the author(s) seem to associate a lot of precedent law and decisions which were based on the "manner of carrying arms" as in "concealed weapons". Some local and state governments created statutes to help eliminate the carrying of concealed weapons which were commonly used by criminals. For the most part, even the founders declared the carrying of concealed weapons to be "immoral" as it was indicative of criminal intent and the element of surprise necessary for such intent. But that didn't preclude law abiding citizens from carrying weapons, which may very well have been concealable, out in the open. Certainly it was never argued by the founders that the criminal carrying of pistols, concealed, should in any way preclude law abiding citizens from possessing pistols on their own property and in their houses.
I think the claims made by the author(s) are stretching the intents of some of these laws in order to make it appear that it has been "common" practice to ban several types of firearms in order to maintain public safety. I'm not sure that's the case, outside of California, New York, Chicago, Washington D.C., and a few other states or localities. Besides, as these folks say the USSC made mistakes, who's to say it wasn't the lower courts who made the mistakes when ruling that certain bans on firearms was "constitutional". That's what this whole process is about. The USSC and the other courts must get to a consistent position on whether the 2nd Amendment offers wide protection of the individual right, and for which types of arms, or if it is merely a warning that government entities shouldn't be too "unreasonable" when they decide to ban certain types of firearms.
I guess we'll see if this attempt gains them any ground. I'm skeptical that it will. They are basically telling the USSC that they made grave mistakes in Heller. While people have done that in the past, has it ever caused a reversal? I'm not sure. Maybe it has.
"If you love wealth better than liberty, the tranquility of servitude better than the animating contest of freedom, go home from us in peace. We ask not your counsels or arms. Crouch down and lick the hands which feed you. May your chains set lightly upon you and may posterity forget that ye were our countrymen." Samuel Adams.