I don't understand some of the pearl clutching surrounding tattoo parlours. The vast majority of them these days, not least due to stringent regulations, are not dark, smoky, dens of iniquity down some back alley frequented by drunken sailors, pirates, ex-cons or gangsters.
It has never been easier to research the facilities, environment and policies of a given establishment... via their own website, customer reviews, general Googling (e.g. for news stories) and it's very likely that if somewhere has a poor reputation for hygiene or customer care that you'll find evidence of this. Conversely you're likely to discover the places that take the responsibility of piercing very seriously. The ones who publish pictures of their studio and past work. The ones who welcome you for a no obligation chat about your requirements. The ones who not only provide full verbal and written aftercare advice but who encourage you to return if you have any problems at all.
In my experience, reputable places usually have a dedicated piercer, rather than someone doing both piercing and tattooing. Not that it would be impossible to be skilled at both but having someone who only ever pierces means they're more practised at it. There is usually a separate piercing area, akin to a doctor's examination room, which is clean and bright.
Having had a number of piercings myself, as well as accompanying my daughter to have her ears pierced twice, the studio we use insert small stud like bars in ears, closed with screw on balls. The bars are very slightly longer than a typical stud earring to allow for any swelling and similarly, ball closures are used to avoid the issue of a butterfly back being pushed too tight and becoming embedded in the ear. It's impossible for a ball to do this as it only screws so far on the thread.
As with anything, if you look hard enough you can always find examples of things going wrong with either method. Not all practitioners are going to be equally skilled or careful. And not all clients will carry out aftercare as they should either. However, the vast majority of issues I've read about seem to be gun related whether it's infections or misplacement. I've never had a gun on my ears but around 30 years ago (before piercing studios were so common) I did have my nose pierced with one on a couple of occasions and those never healed. I originally had my ears pierced with a needle in a jewellers 40 years ago.
As for Claires, as others have said, they're very visible and convenient. And I don't think many people think beyond that. Even if you leave aside the consent issues, I hate the public nature of the process. If your child's already nervous a load of gawping passers-by isn't going to help with that and is highly likely to make it more painful as they'll be even more tense. Ditto the general hub-bub and music. They're hardly a quiet calm environment.