The problem is anyone can own a gun, responsible and fully sane or otherwise.
Of course that's not at all true. Every state in the country makes provisions for keeping crazy and criminal people from buying guns. It doesn't help when one of their family members decides to commit the criminal act of facilitating their acquisition of one anyway. Several states are actively knocking on doors and taking guns away from people who have been convicted of certain crimes, or who have fallen under a protective order or deemed not sane enough to own weapons. You're just (knowingly, I'm sure) wrong on the facts.
Americans need to accept that some people just shouldn't have access to such deadly weapons.
You mean, like we already have? Sure, why not. People who don't realize that just need to check their state laws so they can see that's already the case.
The constitution even says so - you can bare arms as part of a well organized militia, i.e. with appropriate training and checks on who is allowed in.
You're deliberately misrepresenting the second amendment. Even if you can't parse the actual words right in front of you, you can go off and ready countless documents by the people who wrote and ratified that amendment, showing that you've got it exactly, precisely backwards. The founders, having spent years living under the militaristic thumb of the British government in the colonies, were very apprehensive about the continued existence of a standing army (especially a federal one). Still, they knew that there had to be an organized military capacity at one or more organizational levels (at least state, county, etc). Their wording in the second amendment, if you were to use slightly more modern, casual parlance, would go like this: "Because we know there will always have to be a permanent military structure in place to defend the country, we don't want that military to have a monopoly on the ownership of arms, as we experienced under British rule. This amendment officially prohibits the government from preventing the people from keeping and bearing their own arms."
The second amendment was written specifically to preserve your personal right, should you choose to exercise it, to keep and bear arms exactly because there was inevitably going to be a well organized militia operating nearby, and the founders - having seen what they'd seen - considered it absolutely vital that the organized military didn't become the only entity in the country that was armed.
Your fantasy, in which it's exactly the opposite, flies in the face of everything the founders had to say on the subject, and is completely contrary to the debate, writings, and ratification votes that surrounded the amendment's place in the constitution. It's just like the first amendment in that capacity. The first amendment doesn't spell out who's allowed to speak, or indicate that you have to be qualified to own a printing press. It's there to prevent government over-reach, just like the second amendment. And the fourth, etc.