This thread has gone even more nuts 
This isn't a game of top trumps with child beating dog in every category. As far as I can tell nobody is saying that dogs have a greater right to existence or presence in a given area than a child.
And the OP should not have to apologise for the existence of her dog.
The question of "is a blanket rule on dogs being leashed an anti-dog idea?" in my opinion - yes. Not because dogs have greater rights than children, but because they do have some rights to live in a healthy and appropriate manner for their genetic background. I wholeheartedly agree with those saying they're not fur-babies -they're modified wolves (just as we're modified chimps
) and they do need to run around and let of steam. It would be utterly cruel to ban dogs from off lead exercise.
The idea of specialised dog areas would be fine, in principle. Unfortunately I imagine the logistics of this would mean it ended up too small and the high concentration of dogs would have a whole host of other issues - dog on dog aggression, plus it would be very hard for owners to train in such an area without space to work quietly. And there's the issue of compliance - the issue already is idiot dog owners who think it's acceptable for untrained animals to be loose around children. Not to mention those who believe dogs have no rights who might choose to trot across the dog area on a short cut and be similarly outraged at loose dogs
Hey, there's always one! I'm not saying it couldn't work. In urban areas, it may be a big improvement. In rural areas, I still believe dogs should be allowed off lead, under close control on any PROW.
Regarding the assertions that "a dog is not under control unless leashed". This may be the opinions of some on this thread, but under law, the definition is under close control, leashed or will return at first call.
It is not arrogance to have a well trained dog off leash. It's (should be?!) experience and years of training. I am confident my dog will return at first call because I've put that to the test. He has a distance drop and a recall that over-ride his natural instincts. Your child could jump up and down screaming, waving their arms around, having rolled in chocolate ice cream and throwing his favourite toys in the air, and he would drop and wait. I know this, because I've tested it, in controlled environments (ok, never rolled a kid in ice cream, but...) so I can be confident he will comply.
Yes, dogs are unpredictable. So are kids. And adults. Heck even trees can be unpredictable (more than once the sneaky devils have dropped branches on me when I've been walking under them). And as my latin teacher used to yell "BO! the future is always unpredictable"
. I am still willing to bet that my dog ignoring my commands and turning savage is less likely than me turning savage and going on a biting spree...