I think this article might clear up your question on Switzerland's lower gun crime rate with a high private ownership rate.
The article a bit lengthy, but I do believe it's worth the effort for most people to read.
Also keep in mind this did come out back in 2013, so some of the data such as numbers has changed, but ultimately the explanation still holds validity.
Switzerland guns: Living with firearms the Swiss way - BBC News
I knew there were many good reasons why gun deaths were lower in countries like Switzerland and The Netherlands, but after verbally banging my head against the wall in favor of stricter gun regulations, I didn't want to do the necessary search for the answers.
I am not against gun ownership, but I do believe that with so very many guns in the hands of anyone who wants as many as they can get, there is room now for the pendulum to swing the other way and have some sensible gun regulations.
Some here are too young to remember that it wasn't until a right leaning SCOTUS did give some weight to the last part of the second amendment and only in this century codify a right to have guns. Justice Burger, despite his conservative bona fides, did not believe the 2nd amendment created an individual right to possess guns.
On the contrary, he said, "the Second Amendment has been the subject of one of the greatest pieces of fraud, I repeat the word fraud, on the American public by special interest groups that I have ever seen in my lifetime."
Burger, speaking for the majority of judges and lawyers, said the Second Amendment was meant to forbid the federal government from abolishing state militias.
Most people in favor of this forget that the entire amendment reads, "A well regulated militia, being necessary to the security of a free state, the right to keep and bear arms shall not be infringed."
On December 15, 1791, when the Second Amendment was ratified, women could not vote, people were allowed to own other people and guns had to be reloaded after every shot. The Second Amendment, as Justice Burger said, was never meant to create an individual right to own guns, but to forbid the federal government from abolishing state militias.