In the event 38/357 would have been available during the time of the old West, in revolver and rifle both, would it have been more popular, less popular, or the same, as .44-40?
Tough comparison as a lot of things would be different if smokeless powder had been around 50 years earlier. Confining the choices to BP, the cartridge sizes to generate those levels of energy would be rather large. 45 Colt was quite a step up from 44-40 in BP loading but still not close to .357 magnum power. All of the big three rifle/pistol cartridges of that era 32-20, 38-40, and 44-40 also have thin necks that helped seal off BP fouling getting back around the chamber.
Looking at it from a smokeless perspective and using the steel available at the time, revolvers might look more like Century 100s than Colt SAAs.