@Glomoro
The reason I think archery gets a bit of a pass is because kyudo/kyujutsu is so big (for a martial art, anyways, it has something like 200,000+ practitioners) in Japan and so many schools have archery clubs. As far as traditional and cultural stuff goes, it's something that's widely respected. I'm guessing it's similar to things like the tea ceremony or calligraphy or something. And while the sword/katana is also as Japanese as it gets, kyudo is also different in that it's very zen focused. I remember reading one of the kyudo archery masters even chiding how he wished more students and practitioners stopped focusing so much on hitting the target accurately and instead focus on the mental and ceremonial aspect of it, like having the best form possible and clearing their mind - accuracy on target be damned. Because of that, it seems like the bow is viewed as less of a weapon.
As for slingshots, I have no idea if they're regulated, but they don't seem like it they're focused on much. And things like this from "Saving 80,000 Gold Coins" make me wonder:
...Then again her brother also owned folding knives and camouflage, so even as a Japanese person in Japan he would've already been thought of as odd for it. But given how specific the author/mangaka was, it seems more likely that they actually have experience with slingshots rather than just referencing them.
Crossbows have an interesting and varied history, which makes me give pause. Even in the west, some places have strict laws even to this day on crossbows (especially when it comes to hunting, but even just owning them). Crossbows weren't as big in Japan as, say, China or western europe (to the point where Pope Innocent II banned crossbows, as well as slings and bows, against fellow Christians), but I think it would be viewed as a complete weapon of war in Japan, like the sword, and thus regulated.