Sorry to hear your experience. Glad you and your dog are okay.
I get total mother bear syndrome when these incidents occur, but always feel better if I can confront the other person calmly - how on earth you managed to walk away and not say anything at all is beyond me. You have the self-control of a saint.
Fortunately, the dog park in the adjoining suburb to ours is pretty reasonable compared to our local. Most people are good and I have discovered which times to go and hang out with a regular bunch. There is still the odd idiot - but it's much easier to deal with someone with an out of control dog if they are in the minority, lol.
I have also learnt a lot more about subtle signs of trouble between dogs and it's always those dogs who are not distracted and diverted at the right the moment (consistently), that cause problems for everyone else. Some of these owners are able to see their lack of knowledge and it makes everyone much more willing to work with them and their dogs.
Others will remain oblivious to the collective groan that goes around the dog park when they pull up in the carpark.
I totally blame the human in the situation - the dog is invariably being given the wrong training, or no training at all and they are brought down to the dog park so the owner can 'exercise' the dog and be passive about the dog's interactions. Then there are those who will actually get annoyed if someone else intervenes with their dog. Funny how these two characteristics go together......
We have some regulars (dogs) that bark a lot and it does trigger certain other dogs to get confrontational. Those owners who deal with it effectively tend to de-escalate the situation and get on with things. My two youngest pups have had the barking reaction when arriving at the dog park or if a new dog arrives. Like you, I put them on the lead. I give them a verbal correction (quiet but firm, in a low voice) and walk them away a metre or so. The second they are quiet, I praise and when they settle, they are rewarded by coming off the lead. If they persist, they go back on the lead and go for a longer walk in the opposite direction until they realise they are not going to be able to engage when they are barking. We have an older dog who did the same thing initially (she is fearful dominant) and she has also responded to this kind of thing - much more slowly, of course. There is hope for your dog with socialising - they may never be willing or able to get right in there and play with other dogs, but I think it's worth trying to get them a certain way along if you can find sympathetic dog owners with dogs who don't get flustered by your dog barking and stressing.
Hope you don't run into the family under those circumstances. Kids walking dogs should NEVER be allowed to get out of control - it can be upsetting for the child (hence the kid blaming your dog) and teach the dog it can overpower the kid when it wants to.
P.S - I throw my leads at dogs if they get into a tussle (safer than hands and can provide the second of distraction for you to safely grab a collar). I've also been known to yell and throw a few well aimed kicks at the offender. Anything to avoid putting my hands in harms way.