Why can't a first-order block have an oscillatory step response?
It's apparent that a first-order block cannot have an oscillatory step
  response.
  
  How come?


A first order system will reach a state of equilibrium in the fullness of time and the energy transfers between components will also reach equilibrium and remain stable in the fullness of time.

An underdamped second order system will not have the component energy transfers expired when its output apparently reaches equilibrium and, because of this, the output will overshoot. 

As an example, a low pass LC circuit will charge the capacitor up when a step voltage is applied to the input and, when the capacitor voltage equals the magnitude of the step input, there is still energy left in the inductor that will cause the capacitor to carry on being charged to a higher voltage.  Eventually this will die out (in a series of ever-decreasing oscillations) due to circuit losses producing heat and thus getting rid of the excess energy.