Actaully, they can, but the circuit I suggested would not do that. You are right, I did not think about the fact that it must move slowly. A servo will position anywere in between two extremes, proportional to the pulse with. So to move it slowly you would have to slowly increase or descrease the pulse width.
The way to do this without a micro would be to use a voltage controlled constant current source for charging to get the pulse width, and apply a ramp to it. The constant current source is nothing fancier than a common collector transistor with a resistor on the emitter side and the base being the control voltage. But this sitll way more complicated than my first idea.
The ramp could be created by charging, discharging a capacitor.
A micro would be far easier at this point.
Or, an alternate solution is just to use a low RPM gear motor and limit switches.
Bob