I wonder if it's the right to carry guns which in Europe seems not to be the case.
No.
The "Right to Carry" movement has only appeared in the last ten years or so and has not been linked to any increase in crime levels. Individuals with Concealed Carry Permits ("Right to Carry") don't factor into crime statistics as they seem to be an abnormally fastidious law abiding group.
As far as Right to Carry being a factor with your Chicago incident, no, it didn't play there either. Chicago/IL has no right to carry law. You cannot carry, openly or concealed, a firearm in IL. You are breaking many laws in Chicago by having a pistol and carrying it anywhere on your person, not to mention more laws if you discharge it, and even more if you shoot someone with it.
For a brief on the US "Right to Carry" movement you can find it on the internet but in a nutshell it boiled down to this - many states had concealed carry laws on the books. But with a catch - after you'd followed all the rules, taken the proper classes and so on it came down to the local law enforcement head granting you the right. Historically, if you weren't "in" with this person you weren't getting a carry permit. It came down to his/her personal decision. There was no requirement he/she had to issue the permit even if you were the model citizen and met all the requirements specified in the law.
The "Right to Carry" movement cleaned up the language of the concealed carry laws and made the law "Shall Issue" - that is to say, if you met the requirements A, B, C, D, and so on, passed your classes, not a felon, not crazy, etc. they "Shall Issue" the permit. No arbitrary judgment by a law enforcement official deciding if you could or could not have the permit. But IL/Chicago still has no carry provision, that state is particularly "anti-gun".
Sorry for the diversion, now back to your regularly scheduled thread which, by the way, is very interesting.