Explanation of 2nd order transfer function
A zero would would complicate the dynamics and detract from conceptualisation/ROT, as would a higher order denominator. Unfortunately, not all systems can be adequately modelled by a standard 2nd order TF.
Some standard forms include a gain term. A gain term does not affect the shape of the transient response - just the magnitude and steady-state value
The 2nd order inhomogeneous ODE defines or approximates many fundamental engineering systems