WriterofDreams "but I was asking what the solution is, I'm discussing the problem with you."
Um....I didn't get that at all from your OP; it didn't sound to me like you were asking what the solution is, it sounded to me like you just wanted to have an anti-dog rant, and have people agree with you. 
I agree with what lots of other people have pointed out; anyone who holds up Ireland as a paragon of dog ownership is either ignorant of the facts, or has little interest in the welfare of dogs. I assumed you were the former, but then you stated that you know Ireland has problems with dog welfare. I totally fail to see why you would then hold this up as a preferable situation here.
I think YANBU to be annoyed at dogs jumping up at you; it is completely unacceptable. A dog merely approaching you, however, is another matter, imo. If you are scared of dogs, then I appreciate that a dog running near you or approaching you (without touching you) many be unpleasant for you, but I think that that is something you need to master yourself, and well before your fear starts to rub off on your DS. Maybe some CBT if it bothers you so much? The fact is that we share our environment with dogs, and whilst owners have a moral and legal obligation to keep their dogs under control, there are people who can't or won't.
Most dogs that approach you mean you no harm, and are just curious or friendly. If you ignore them, they'll go away! Your comments
"On every single occasion, often more than once, a large dog has come up to me, often running."
"On a few occasions I have been scared witless by dogs viciously fighting on the path."
make me think that your fear is leading you to misinterpret the situation, seeing threat when there is none. A dog 'coming up to you running' may well mean that he's just going to run right on past you - you are completely incidental to him. My small spaniel goes hurtling along on the scent trail of whatever, and couldn't care less about the people she passes. If you are scared of dogs, dogs playing with each other look like they're fighting.
If your fear is curtailing your enjoyment of the park, either do something about your fear, or find another park. Yes dog owners should have their dogs under control at all times, but if you consider them running past you and playing on the path to be out of control, then I it's you who is the problem.