However, disproving that doesn't prove the opposite, i.e. that mass gun ownership reduces gun deaths or stops crime.
No, it doesn't, and I never said it did. I was just attacking a common talking point for the anti-gun crowd.
Their gun related suicide rate is one of the highest in Europe.
Well that's no surprise. I'm surprised actually that it isn't the highest in Europe. Maybe the good economy (versus places in Eastern Europe) makes less people suicidal. But is the rest of Europe suicide-free? What are the rates, after you add together both successful and unsuccessful suicides? Having a gun available just makes it more likely you'll succeed; other methods aren't generally as sure-fire (pardon the pun).
One common argument made for gun ownership is criminals such as burglars won't break into a house with an armed owner yet burglaries are rising in Switzerland.
Burglars, by definition, are people who avoid confrontation. They look for patterns, to see when a house is unoccupied, and break in then so they can steal loot. The people steal face-to-face are called "robbers", or possibly "home invaders". Are there a lot of home invasions in Switzerland? I suspect not. Lots of places have relatively high property crime rates (or just petty crime), with very low violent crime rates.