The point of this particular thread is that the sensitivity of the two green lanes within the Peak district is worrying for the safety of the children, horse riders walkers etc, they are put into a dangerous position!
It's only in relatively modern history have they become recreational routes and with that recreational use, comes conflict. In fairness I have to say I have had no experience of 4X4's and so don't know how much of a problem they are to other users, but I have almost been hit by mountain cyclists a few times, because I wasn't paying attention. My first reaction was, "shouldn't be allowed", but I'm not sure that's right, and in fact I think it's wrong. The reality is that the route I was on is used by horse riders, cyclists, joggers, runners, walkers, strollers and people just having a wander, and all should be on the look out for the other.
I do believe pretty strongly that if you tell one group of people, that you can't do it , end off, period, then they will react and organise. There are millions of people who don't care if they drive off road or not and hundreds of thousands with 4X4's that might feel inclined to give it a go.