I use them in 2 instances when they can do the job better than you -
On all citys I have "emphasise production" on. This means that the turn the city grows, when it cannot store any more food anyway, it has the highest production possible. This is especially important for setler factorys.
When I have captured a city, it usually has close to all the citizens revolting. Even if you set any that are not revolting to entertainers, then the city may riot next turn if you quell many people. If you get the guvnor to manage moods in the city, it will not.