Why does an elevator deflection (step) result in a nonzero short-term-steady-state response in pitch rate?
The equation is certainly oversimplified, hence the name 'short period': it only models short period pitch oscillations. In this model, there are only two states: pitch and pitch rate. Any other effect is thus neglected.

A step response in an elevator would in the short term definitely lead to a nonzero pitch rate, as the model predicts. In reality, the pitch rate will usually decay to zero, because the airspeed drops to zero (there may also be altitude density effects). Imagine what would happen if you would pull on the stick in an aircraft, and crank the throttle to make sure the speed does not decay. You would do a vertical loop, exactly like the $q$ term predicts.

Of course, then you run in to an extra complication: your model is linearized. The concept of large angles, let alone a loop, does not exist in the linearized world. Think of a linearized pendulum: that works fine for small angles, but as far as the model is concerned, a 360° angle means just a very large deflection, and not the reality of returning to zero deflection after a full loop. The model is then only valid for the small amplitude oscillations it is intended to predict.

A phugoid mode exchanges speed for altitude and vice versa. Neither of these states are represented in  your short period linearized model. This behaviour is impossible to see in your model. You might as well ask why you don't see the effect of the Moon position in your model; it's simply not in the state space!

A long story short then: simplified models like these are great tools, but cannot and should not be used outside of what they are supposed to do. If you want to estimate instantaneous pitch rate from elevator deflection then this is the model for you. If you want to check for trim conditions, large attitude deviations, or even roll behaviour, then you need a better model.