For the OP, what kind of advice are you looking for? Or are you just looking for validation?
Some thoughts I had reading this thread.
Not everyone shares the same world view and the same ideas of parenting, and certainly we don’t all agree on how much other adults should or shouldn’t correct other people’s kids.
One major difference in worldviews is the degree that “rules” need to be followed. At three, emphasizing them is better, but not all parents do that all the time, and the older the child, the more flexibility (some? most?) parents give.
It’s really important to keep in mind what is age appropriate. Three-year-olds need constant reminders to not bump each other off ladders. Seven-year-olds need to get bloody noses to learn the limits. (IMHO, of course, other parents will disagree.)
Obviously, as a parent, it’s imperative to protect your three-year-old from older kids who play roughly either by removing your child or getting the other child to stop, either through talking to the other parent or by saying something directly.
When you have to talk to someone else’s kid, it’s going to be much smoother if you follow the advice given by others in this thread by being polite and reminding them to be careful. Treating children with respect by telling them why their actions could hurt someone else goes a long ways to not escalating the situation. This doesn’t mean that you can’t be firm or that showing respect makes you lose somehow.
Lefthand of Dorkness’s post is really good. Rather than jump on people for not following the rules, teaching kids why the rules make sense is helps them grow.
Not all parent agree. My parents were both very authoritarian, we had to follow rules simply because they were rules and there were no questions allowed. I don’t like that myself, but some people do.
With confrontations in public, stressing safety first makes your position stronger. People are going to have a hard time arguing against that. The second level is appeals to fairness, such as if kids are cutting in line. Again, if you’ve told an older kid to not cut in line, their parent would have to be a real asshole to argue that OK.
As you aren’t the Sheriff of the Playground, placing the argument entirely on “The Rules” is pretty weak, especially if the rules aren’t necessarily universally followed. You may think they should be, but not everyone does.
For a bunch of rough and tumble seven- and eight- year olds, their activities aren’t seen as “misbehaving” by many parents if there aren’t other kids around.
You said yourself that you were pissed off, which is going to lead to confrontations. Challenging someone’s parenting skills based on your values will get make some people unhappy.
If you wanted to deescalate the situation, rather than you comment: “Look, that kid was horsing around, he was keeping my son from playing, he’s not even supposed to do that, I stopped him, there’s no problem here,” if can be framed as appeals to safety and fairness. “That kid went up the slide right when my son was going to go down, so I was concerned about his safety.”
I’m not really sure what you mean by “keeping my son from playing.” Is it that he’s “jumping” the line? “Hey, can you wait until he comes down?” can address that.