All sounds nice in paper, but yet, you still can buy cocaine easily, and the laws on it are pretty much just as rough as the ones you just proposed.
Obviously there would a good benefit on performing a good background check and actually training the person how to safely handle the gun instead of just "LETS GIVE WEAPONS TO EVERYONE LOL", but banning guns won't solve the issue of people that want to commit crimes having access to weapons.
The best approach is giving people less reasons to want to commit crimes in first place.
You are trying to avoid the crucial issue here. The problem is not whether we can go to zero crime, because... Newsflash! There is crime, even in countries where there are very strict gun laws.
Here is an example: Japan. Lowest gun crime rate in the world. Ultra-strict laws regarding gun ownership. And Japan is also a country where organized crime, the Yakuzas, is practically out in the open.
The goal of banning guns is to ban guns, and reduce gun-related violence and homicide. Banning guns will reduce violence and crime acros the board, but it will not suppress it completely.
Let's take your example: in any big city, american or not, you can probably find out where to buy cocaine. The difference is that, in the US, an addict can pull a gun on his dealer (or the reverse) and kill him for his stash. And whoever survives can then use the same gun to shoot at police officers and kill them. In the rest of the world, while this can happen, gun violence, even between addicts and dealers, is a lot less prevalent.
Yes, crime and law-breaking will always exist, gun ban or not. But banning guns will reduce violence and, ultimately, the number of deaths per year.
Here is another example: if the Florida shooter did not have access to guns, and he went into that disco to, say, stab people with a sharp knife or sword, he may have been able to kill a couple of people, at best, before being subdued by bouncers and others in the disco. This would now be treated as a horrible hate crime, and not as the bloodbath -- 50 dead, scores wounded -- that it is.
The Economist said it best years ago when they said "It takes a lot of knowledg and strength to kill someone with a knife or a blade. It takes none to pull a trigger." (and I paraphrase from memory).
Ban guns. This is the best solution overall, and it requires nothing more than courage and a willingness to tackle this problem.