My understanding is that the device allows the firearm, including the trigger, to recoil back far enough that the finger that was on the trigger will release it; the spring then pushes the firearm back forward such that if the user's finger hasn't moved it will pull the trigger again.
If the makers of the Atkins Accelerator wanted to push the issue, they should move to the district where the Rock Island Armory case was decided (which struck down the 1986 ban on full autos), register an Accellerator, pay the $200, and then sue for a refund of that $200. That's essentially the approach used by Thompson/Center Arms to get a Supreme Court ruling that a kit containing a Contender pistol frame, a stock, an 18" barrel, a 14" barrel, and a note warning against attaching the 14" barrel and stock simultaneously, was not a "short-barreled rifle".
I think anyone that bought one could make the ATF buy it and pay their attorney's fees and damages. You are correct about the Thompson center, but the case that ATF was trying to make was that attaching the barrel that would chamber and shoot a 410 shot gun shell would be a sawed off shot gun. The supremes laughed at them.