The only difference here is you make the appointment in person.
It's really not a hard concept to grasp but after reading your other comment about how you love to talk confidently about subjects you no nothing about, I'm starting to question if this is one of those subjects. Perhaps you don't even have kids at all and this is just another one of your attempts at trolling which you also described enjoying.
In any case, the real crux of the matter isn't what you consider logical but rather what the generally agreed social etiquette is. So you can argue until you're blue in the face but if it's not how barbers currently work then that simply isn't how barbers currently work. Period.
We’ll just have to end it here.
It's relevent because this topic is about parenting and kids waiting in line for stuff like barbers.
> I've also never known a barber that didn’t take an appointment.
As I've repeatedly said, this might be a cultural thing. Most UK barbers in UK towns don't take appointments. Trendy ones in the big cities might well do, but in terms of your typical UK barbers shop you would just turn up and wait in line. So to that effect, I've also never known a barber that does take an appointment.
> We’ll just have to end it here.
For what it's worth, I did try to end this tangent several times already when I compromised with "it might be a cultural thing". Which, weirdly, is when you decided to rejoin the discussion. Go figure.
Sometimes people do disappear when it is extremely busy but when they come back (typically an hour or two later) they expect to join the back of the queue. Literally the only time I've seen anyone leave queue and rejoined it where they left was when they were there with their larger family and one of the kids urgently needed the toilet. And even then I've only seen that happen once in the decades (literally!) I've been frequenting barbers.
You can use all of the patronising terms you like to promote your preferred etiquette but if it's not shared with the rest of community then your method simply isn't the "proper way to do it".