There is more to it than that. First of all, he did not only say that the execution was to be administered by the Islamic government, he also reminds us that Islamic Shariah has a due process of law in which specific conditions ought to be met before any capital or major punishment (hadd) can be carried out. Vigilantism is not permissible by Islamic Shariah. Most of these news items that you hear about such crimes come from the actions of vigilantes.

Contrary to what may be assumed, there were not many people during the time of the Prophet Muhammad who were executed for adultery. Islam is a religion that emphasizes on eliminating as much evil as possible from the public space. If you are going to commit a sin, then at least do it in private. In order for any man or woman to be convicted for adultery, there have to be four witnesses who have a good standing within the community who testify that they witnessed the act of penetration, and furthermore the sentence cannot be carried out if suspicions are present. It is extremely improbable for four witnesses to see the act of penetration unless you were a person who had no shame and brought your sins into the public space, in which case you have nobody but yourself to blame. Allah likes to forgive the one who conceals his misdeeds. Allah forgives all sins.