Why not?
Wouldn’t a better solution be for parents to monitor their children’s internet activities just like they watch them in a public park or at a mall?
That would be an additional solution. However, parents can't monitor their children 100% of the time. Some parents are negligent, and the children shouldn't suffer because of that. Even if the parents are not negligent, some children do very foolish things, and unless you lock your children up, they are going to be unsupervised some of the time.
Suggesting that a campaign to encourage parents to monitor their children better will address the problem is extremely naive.
What is wrong with private individuals using their own resources, to take positive and legal steps to identify pedophiles and collect evidence that can put them in jail?
That's not what Dateline NBC was doing in Texas. If that is what they were trying to do, they failed miserably. They are Zero-for-Twenty on prosecutions, let alone convictions. All they have to show for it is one dead guy.