They already said they're European. The terms you mentioned are very Americanized/America-centric phrases.
The rest of the world doesn't really have the same concern or context, because the rest of the world doesn't have the same issues or political/media environment.
If you buy a gun from someone on a classifieds site or friend/family, in most states you don't have to get a background check to transfer ownership of the firearm. This is typically the "loophole" that people refer to when they want Universal Background Checks.
I'm not sure what you mean by buying illegally - whether you mean to someone prohibited from owning firearms (like a felon or something), or buying "illegal" guns... Either way there are usually stiff penalties for owning restricted devices. For example if I were to 3D print an auto-sear for an AR-15 (making it full-auto), that's a ticket to a 10 year sentence in federal prison. Assuming I don't have the permission slip from the ATF. As for prohibited persons buying a firearm, I'm not sure what the penalties are, but at the least it'd be a violation of their release?
Buying legally in the US isn't that hard to make it worth it to buy illegally unless you are a broke teenager buying one to try and "look cool and tough" to your friends.
in most states someone who has multiple assault violations and a history of mental health problems can walk into a gun show with $700 and 15 minutes later walk out with a semi-automatic AR-15
That's not an assault rifle, first and foremost. Second, selling to someone with multiple assault convictions is itself a Federal felony, and not every gun owner is so careless as to want to take that risk and get pilloried in the press. You're generalizing a worst-case scenario based on a stereotype.
it's not a worst-case scenario, it's a very obvious loophole that should be closed — private gun purchases should require waiting times and background checks
"not technically an assault rifle" doesn't matter in the slightest, no one's impressed by this constant pedantry... yes, yes, it's an ArmaLite 15 and isn't technically an assault rifle because it's only semi-automatic. This changes absolutely nothing, mass shooters are essentially cosplaying during mass shootings with this thing because it looks like an assault rifle and it's easily accessible.
the proliferation of modern small arms is one of the worst things to happen in human history
the facts being "looks like an assault rifle but isn't technically an assault rifle"? that's immaterial to the point, and it's clear you have no desire to engage in a good faith conversation, see ya
The rest of the world doesn't really have the same concern or context, because the rest of the world doesn't have the same issues or political/media environment.