If ALL dogs should be on a lead because SOME dogs are out of control, does that mean that ALL teenagers shouldn't be allowed in the park unless accompanied by an adult because SOME of them cause a nuisance; smoking should be banned from parks because SOME people drop cigarette butts on the floor. ALL children should be kept on very short reins because SOME of them try to kick the pigeons which are there. ALL children should be kept on reins because some of them come up to me when I don't want them to.
And of course I appreciate that the situations are somewhat different because a dog could possibly bite, but the majority of people, including dog owners are decent, considerate people. As described above, there are other people who are not considerate but that doesn't mean that all people (or dogs) potentially falling into that group should be punished. Most dog owners are responsible, they will clean up after their dog and will keep their dog under control. Lets not pretend that there are packs of marauding dogs causing havoc round the park, just waiting to pounce on the children. Most of them prefer playing with a ball and each other and aren't really interested in children.
I believe that the play areas should be dog free, and the majority are, but that dogs should be allowed to run free in parks rather than having to be on leads. If people cannot control their dog, they shouldn't be letting them off lead; there is already legislation in place to deal with out of control dogs and this should be enforced rather than bringing in dog control orders to every park which will punish all dogs, not just those that are out of control.
And yes OP, that sign would have annoyed me (a little). It's unlikely that the child would have written it and put it up without parental assistance. The parent should try explaining to their child that dogs do not have to be on lead in that area, and it is not up to them to try to persuade otherwise. I think the dog mess sign was a good idea though.