Why do we use step response?
Step response and transfer function are interchangeable via Laplace transform (step response -> differentiate to get impulse response -> laplace -> transfer function), but you can't run a Laplace transform on a graph pulled out of a datasheet...

For example, the user of a LDO will usually be interested in things like voltage sag/overshoot when load current steps up and down, so in this context a graph of the step response is useful.

Same if you design the control system for something like a mechanical actuator: the transient response is often more important than transfer function, you'd want to avoid overshoot, avoid excessive slew rate, have good damping, things like that. Step response shows all this in a way that is easy to understand, whereas transfer function will not (instead, it provides more insight into stability, etc).

Also testing the step response is much easier than measuring the transfer function.

And the transfer function is only valid when the system is linear, not when it is slewing. If slew limit is involved, or other large signal conditions, then step response is no longer the Laplace transform of the transfer function, because the system is no longer linear time invariant.

Opamps' datasheets usually give both, since we're both interested in the frequency response, and settling time/ringing/overshoot, clipping recovery, etc.

EDIT: I just realized you asked why step response was used instead of impulse response:


It is easy to generate a fast step, very difficult to generate an "infinitely short" pulse (or an approximation of).
The step contains much more energy and will generate a much larger response. Whereas a pulse would only generate a small blip on the output. Signal-to-noise ratio is much better with a step.
An impulse would not allow to test for slew rate. It would provide little information.
Thus the step is easier to use in practice. If you want the impulse response, do a step, then differentiate.