The design would have to incorporate some sort of device to support the "rim" as the firing pin strikes it, yet doesn't interfere with extraction.
The extractor claw grabs the rim from the front like it usually does. And the firing pin strikes it from the back, crushing the rim between the firing pin and extractor.
Though the biggest upside of 22LR is the cheap ammo. With the cost of buying a new rifle and some newly designed niche ammo, you may as well just get some common, low-caliber centerfire.