The Guardian article is very interesting on the various parties with their own axes to grind getting involved with these cases, but it ducks out of discussing the problems with mediation, the main one of which is that both parties have to agree to take part.
Mediation requires both parties to agree to take part, be open to alternative solutions and other ideas. I am not convinced that in cases like the current one and others mentioned in that article, that there was ever any possibility that the parents would deviate from their insistence that the child should not be allowed to die and life support equipment should not be removed, and I think cases like these, would still end up in the courts.
This is not to say that mediation would never work, I am sure it would with less extreme cases, the ones that never make it to court anyway. But I suspect the ones we know about, the extreme ones that hit the courts, would still end up there.