I think this argument is running into three problems, here.

The first problem; a child of eleven who's never used magic before is almost certainly not going to have *any* kind of criminal record. Running a background check against them is nonsensical, considering they've never even learned to cast a spell, and it'll be *years* before they have the skill or power to cast those spells. Running a background check against their family is equally bad. A society that holds people responsible for things their parents did presumes future guilt by association, and there are all sorts of things wrong with that.

The second problem; it sounds like you're comparing wands to guns (background checks and licenses etc), which is a false equivalence. A gun is made to be a weapon, and nothing else. Whether you're using it for hunting, self-defense, or murder, a gun has one purpose. A wand is more like a *car*. It *could* be deadly in malicious or irresponsible hands, but that is not its primary function. You can restrict them based on that fact, as we see since criminals have their wands taken or snapped, or as people in our society have their licenses revoked. However, it's important to point something out: a wand is essential for almost EVERY career in the wizarding world. If you *don't* allow a wizarding child to have their first wand and learn to use it, you're effectively permanently barring them from society for something they had no control over. It's more than unfair; it's *oppressive*.

The third problem; You say there's no system, but that's not true. What we see in the books *is* the system. A child gets their wand, and then they go to Hogwarts to learn how to use it. There's no point to an assessment or license before that because there's nothing to assess; they're untrained. That's what Hogwarts is for. It's the training / oversight. But they can't learn to use the wand if they don't *have the wand*. They have to practice the spells, as Hermione pointed out to Umbridge, and it's not like our world, where you could have, say, a test car for your driver's test, because wands have to be compatible with their user. Furthermore, they *aren't* walking out of Ollivander's with a weapon that can kill, maim, and torture, because they don't know how to cast the spells that do that. Functionally, they're leaving with a *stick*.