My understanding (FWIW) is that blowback actions cannot be used with cartridges more powerful than 380 ACP/9MM Makarov because you either need a recoil spring that is so strong most mortals cannot withdraw the slide or you need a slide/breechblock that is so heavy that you would have a 7 pound handgun. With a shoulder arm weight is not as much of a consideration. The Thompson originally had the Blish Lock, later models were straight blowback when it was found that was simpler to maufacture and just as effeicient. Other well know submachine guns-the Sten, the MP38/40, the M3/M3A1, the PPSh and PPS are all blowback actions. The Marlin Camp Carbines have blowback actions.
I have zero experience with the .357 SIG, it seems to be a "cult" cartridge, like the .41 Magnum-which I like. As such it has a devoted-but small-following. Try finding a Marlin M1894 in 41 Magnum.
Since the .357 Sig is a necked down 40 S&W a carbine using interchangeable barrels-like the SIG SHR-makes sense to me. However we all know what a best seller the SIG SHR 970 was.