That hasn't happened for the most part. A LOT of people in the US have guns, in their homes and on their persons. Yet the scenario you have described hasn't come to pass.
Of course the kind of scenario I sketched - a mid-range violent crime directed at property, not at killing or maiming someone, ramping up into the crooks killing their victims when they think they are running into some hard resistance, because hard resistance typically implies guns, or even because the crook/s just panicked - doesn't happen all the time. If that happened every other day in most cities, America would break down completely. But it does
happen with a fair bit of regularity: anyone can read in the news about break-ins, store hold-ups and bank robberies that ended in the conmen killing one or more of the folks they encountered - and I don't mean among the cops. In northern Europe, this kind of thing is ultra rare, and a key reason for this is the frequency of privately owned guns is low and law-abiding people don't typically have a loaded gun in the bedroom drawer or at the shop counter. Though they may have, but it's not expected.
I'm not saying "if you keep a gun at home then it's your own fault you might land in a fight to the death with an intruder and get killed" - rather, if many people have guns at home and everyone knows it's easy to get one or more guns, then criminals will act on what they know about general habits. Even with people who don't have a gun.