Interesting, that makes me wonder if my paternal grandfather ever discussed the Estes fiasco with LBJ, I'm sure they probably did. I wish he was alive so I could ask him. He was pretty good friends with LBJ back in his post-presidency days.
I think, certainly, that some offenders are just as you describe - incorrigible. I've known a few in my lifetime as well.
I don't think we (the US) engage in any kind of serious rehabilitation efforts and that colors people's perspectives on rehabilitation as a possibility in general.
I think I'm certainly ok with rapists/child molesters/murderers getting a single strike, but I certainly don't agree with your attempt to conflate child and cop killers. There is basically never a reason to kill a child, there could be any number of valid reasons to kill a cop, unless you believe that cops are infallible arbiters of justice for some weird reason. The circumstances, in that case, do actually matter.