I'm not arguing one way or another but I want to clear up some very common misconceptions about US gun laws.

in some states, you can get semi-automatic weapons (which are completely banned in Austria) in a shop in just minutes. And that without any background checks, psychological reports, justifications, approval required, without anything like that

This is just blatantly untrue and I wish people would stop parroting it. If you go to any shop you need to pass a federal background check to buy any non-vintage firearm (pre-1899..not exactly a ton of those floating around). The exception here is private firearm sales, i.e. I go to Craigslist and sell a rifle or handgun. The law states the seller has to have no reasonable cause to believe they would be an unlawful possessor (weak, yes). With that said, almost half of the states (22 per Wikipedia) have implemented state-level laws requiring a background check for private sales.

In many states even convicted criminals can get guns like that.

Again, objectively untrue. You are not buying a firearm from any legal, licensed dealer in the US without going through a background check. And a violent criminal offense will get you barred from purchasing. For the 28 states without laws around private sales, the seller can be federally legally liable if they sell to someone that is not legally allowed to have a gun and they use it to commit crimes.

In the US, guns are sometimes a presents for kids which they can just…own and use (while in Austria everything is obviously 18+).

No, a child cannot legally own a firearm. The parent can purchase and own a firearm that they are allowed to use, but they do not own it. In many states if the child hurts themselves or others with such a firearm the parents will be held liable, many states have laws around safely storing firearms when children are around.

While in the US (in many states), you can just carry any gun around in public whatsoever. So even if the police sees you having weapons in public, they can’t / don’t do anything about that

In most states if you don't have a license to conceal carry and you do you are breaking the law and can be charged. I'll say this one isn't entirely false but heavily depends on your state.

A large part of why this issue gets nowhere is that neither side can even agree on what is true today, rather than what should be true to bring down the issue of violent crime. If one side says "They're totally unregulated you can just buy one off Amazon and start blasting. We have to do something!" The other side is gonna think "Well they obviously have no idea what they're talking about, no point in listening to what they have to say"