I never played it either, I grew up in a city, we played in the street. I agree with ArmchairTraveller I wouldn't knock on anyone's door and try and run away before they opened the door because the thrill of being caught never appealed to me but this was the only reason other children played it.
Same. I was always really afraid of getting into trouble as a child, so it would have made my stomach churn to risk it. Think it served the same purpose as a rollercoaster to other kids, though. And yeah, my mum would have been incensed if I'd done it, but I remember her going over when other kids did similar and their mother going ballistic to her, so it's funny to hear reminiscing about the good old days when parents always sided with other adults... not in Sarf Lundun in the early 80s they didn't!
Personally, I think it'd be really unfair if someone who'd simply ensured a child's parents found out what he'd been up to faced any legal consequences. She didn't actually hurt or threaten him, did she (?), just held him responsible to his own parents, which is what as a parent I would actually prefer. But I wouldn't risk doing it myself, partly because I don't fancy failing an enhanced DBS check if it got reported, and partly because a lot of parents would go absolutely nuts if you did - people can get astonishingly aggressive over their children, no matter the actual rights and wrongs. I think I'd probably do as a previous poster's parents did, and involve their school. Scary and embarrassing for the kids, and no risk to me. The story about the head opening the door to the knockees was brilliant. 
The woman could argue that it is preventing them from continuing with the harassment - after all, how does she or anyone else know that they don't intend to come back and do it again?
No, that wouldn't fly. Citizen's arrests are for situations when it's not practicable to seek to prevent/interrupt an indictable crime by involving the police. It isn't a vigilante's charter. Here, nothing prevented her calling the police, and anyway, it wasn't an indictable crime.
Harassment is an indictable offence.
Nope. Aggravated harassment is an either way offence (can be magistrate or Crown) but this isn't aggravated - it's not done to put someone in fear of violence, or to racially harass, let alone to stalk. It's just annoying, so a solely summary offence. Plus you'd have to be able to prove that the child in question had done this before, as opposed to others in the gang, which obviously you couldn't. Harassment isn't a one-off, by statutory definition.
whether all of that would be accepted in a court of law is largely irrelevant - the fact is that the chances of the police charging this woman with assault and battery given the possibility of these defences being raised and the likely views of any district judge or jury are, realistically, zero.
But it's not up to the police as to whether to charge or not, is it? That's the CPS' role. I'd be astonished if a judge or jury were involved, too - it would only be a magistrate's matter, surely, if it did get that far? And a bloke in Dundee pleaded guilty of assault on a child just last week for very similar actions, actually - grabbed the child's wrist and frogmarched them -obviously the law differs in Scotland, but it's still in the same ballpark when the argument isn't about law, but attitudes. (In fairness he knew the child, so there may be a lot more to it which hasn't been reported). I just don't think any of us can say this couldn't ever have legal consequences, based solely on what we personally think is right/fair. It doesn't work like that.
The problem with the law is you have to have a clearcut set of rules, and you can't say, "you can grab someone by the wrist, hold them against their will, and march them where you want if because you are angry about something they've done before, and that will be okay." Now and then you read stories about employers who do it when they find out a staffer has been stealing or something, and there is always outrage that the employer is charged over it. What's tricky here is that we don't regard kids as having the same rights because (reasonably) we have to discipline them ourselves, so we are used to the idea. But under the law, only parents are really allowed that wide margin of discretion in curtailing their rights - even teachers, nursery workers etc have their hands tied to some extent. The idea that a child has less legal protection than an adult when a stranger is dealing with them is dicey, I think. You aren't meant to take the law into your own hands, and there's no specific defence available to a stranger in terms of punishing a child.
Anyway. OP seems to have calmed down on the protective instincts for her son, and I don't blame her for being upset about what she's been faced with here. Some of the posts have been horrible.