If you have complete control of your dog, it's easy to avoid confrontation from other dogs providing yours is fast enough.

I'm much better at reading other dogs than mine is (he thinks snarling teeth and fur is an invitation to sniff) and his recall in the park is very strong. I can also send him away (at the moment with use of an emergency tennis ball in my pocket) so without him knowing it, I have control over who he does and doesn't sniff. In parks it's also easy to see them coming before they're close enough for an investigation and I can wander in another direction. I have no problem with other people's out of control dogs because it gives both me and Shadow more experience. If I had no control over my own, then no, other out of control dogs would be terrifying at times but that's as much my fault as their owner's.

If I enter a park and there are other dogs playing, do I have a right to order them to leash them so I can unleash mine? Not at all! and that's the point of the OP. The rudeness for that lady to lay complete claim to the park no matter what her judgement is of a 10 year old or excitable puppy, if she isn't happy letting her dog off leash in their presence then she shouldn't let her dog off leash in their presence. She has no right to order him to do it so I don't blame you for being angry.

That's definitely a difficult one. Both parties would have to follow the same rules for that to work. If one does, the dog wanders over of it's own accord and any training you've put into place is gone to pot.

Dogs are animals not cars. If your car shunts the back of someone elses, you had complete control. If a dog sniffs another's bottom, they had control not you. I appreciate legally it's a different matter, but common sense says that innocent, natural behaviour such as this happens and cannot be avoided. What can be avoided is fighting, but to date this has never happened with mine.