This sounds a lot like Japan. The laws against everyday citizens owning weapons in Japan are very old. Basically, lords and their samurai were able to have weapons, but others could only be armed at the lords' discretion. Guns (and swords) are generally illegal in Japan now. Their level of gun ownership is miniscule. Even police officers don't take their guns home. They leave them at the police station.
Japan has very little gun violence. Part of this is simply that they have no guns. Part of this is that simply carrying a gun (or bullets) is illegal under almost all circumstances. Part of this is their lack of an exclusionary rule. In Japan, you can't argue that a search was invalid and have the results thrown out. You can sue for an invalid search, but if the search is successful, then the search is generally considered valid. Regardless, if evidence is found, it can be used.
There are four reasons to imprison someone:
- Incapacitation. To keep the person from committing more crimes.
- Retribution. To keep the friends and relatives from collecting their own justice.
- Deterrence. To keep others from committing the same crime.
- Rehabilitation. To keep this person from committing more crimes after release.
The death penalty only provides the first three. It's quite effective at incapacitation and retribution. But then so is imprisonment. It may be mildly better at deterrence--results are mixed. It's utterly ineffective at rehabilitation, but the incapacitation improvement may be enough to overcome that.
Neither imprisonment nor execution are particularly good at deterrence unless the prospective perpetrator expects to get caught. In fact, no penalty is effective unless perpetrators expect to get caught. People don't decide to commit crimes because they are willing to accept the penalty. Criminals expect to get away with it.
The truth is that even small penalties like a few months in jail have most of the deterrence value of longer sentences. So the truth is that this society would probably be a lot like Japan. Yes, harsher sentences, but this won't matter so much. Convicted criminals will be less likely to engage in more crimes because of incapacitation rather than through psychological effects, but only slightly less likely.
If you want to deter crimes, you need to catch more of them. This is one of the bases of "broken windows" policing. If you don't think that you can get away with small crimes, then you don't take the bigger risks of larger crimes. So if you want to deter people from committing crimes, change their perspective on the likeliness of being caught. And the easiest way to do that is to catch them.
In theory, harsher penalties could compensate for ineffective prosecution. But the problem is that to the criminal, a 30% chance of the death penalty is not worse than a 100% chance of a one year sentence. And even if the criminal has a 30% chance of getting caught, the criminal will think that it's more like 3%.
Another issue is that you want to deprive the criminal of the benefits of crime. Not so difficult with theft. Just take back what was stolen. More difficult with violent crimes.
There would be very few crimes if there was 0% chance of benefiting and 100% chance of punishment. This is almost regardless of what the punishment is.
Another interesting question is how much it would matter if this country did have an exclusionary rule. With an exclusionary rule, it is much easier to carry contraband, including guns. If Japan introduced an exclusionary rule, would criminals smuggle guns as well as other contraband (e.g. drugs)? How long would it take to get a society more like Chicago where few law abiding citizens have guns but many criminals do?
I would expect this society to be a lot like Japan. The harsher punishments would have little effect. Japan already has these kinds of gun laws and very little gun crime. So long as the laws are enforced similarly (e.g. no exclusionary rule), the results should be similar.