They don't get publicly killed because it does not serve the greater good -- society.
How can a government condemn murder, yet inflict it itself?
Firstly, people will grow up watching these so called "public killings" and truly start to believe that it is right, as you are saying it is, to kill people who fall outside certain social norms. You really think this will bring balance, peace and justice to a society?
Secondly, paedophilia is a social construct: the legal age in England is often different to that of other countries. People are biologically mature at a fairly young age. The state, however, decides that people are mentally mature enough to have sex on their 16th birthday. Again, this isn't the case in all countries. Values and norms are constant over neither place nor time.
While some cases are blatantly just wrong, making "exceptions" and publicly killing them must be equally as wrong. Not from a moral standpoint, but a pragmatic standpoint: it just doesn't work.