I cringe when I hear people describe the way piercings were got back in the old days... usually these stories also come with sidenotes of infections or other complications they suffered through before they healed.
A professional piercing shop will often have age limits (what the limit is depends on the country/area) either because it is required by law or because they don't want to battle angry parents 10 times a week because christian soccer mom Karen thinks 14 year old Betty's new snakebites will land her in hell.
The reason a piercing at home might be problematic even when done by a professional piercer is that it's not a professionally sanitized environment. This can raise the risk of infection, but I doubt it would be much worse than having your kid get their ear pierced at Claire's.
As for claires and such, piercing guns are better now than they used to be from what I've heard, however they are still clunky, violent (the way they pierce the ear is with a force that causes more trauma to the skin and thus negatively impact the healing and quality of the hole) and above all else: they can not be as thoroughly cleaned as a piercing needle!!!!! This raises the risk of transferring disease from one customer to another, NOT a fun ordeal.
This being said... 6 out of my 10 piercing holes were made with a piercing gun, back when I didn't know better. While I did have a small issue healing one of them, they're completely fine today. It should also be noted that ANY bodymod comes with risks.
But so does going for a car ride...
Ngl tho still piercing needles > piercing guns, in terms of experience (and like... everything else). Kids who want piercings suffer way more than they need to just 'cause bodymods and proper piercing studios are still taboo...
ALSO!!! If you are lucky with the pro piercer, you'll get someone that offers numbing cream. Does little difference on ears, but can help ease the anxiety for a nervous kid. (Numbing cream makes MAJOR difference on oral piercings tho)
So that's a thing you could add to the story if you want. Numbing cream should sit for 30-45 minutes, err on the longer side for ears.