Why impulse response
Having the impulse response allows one to simplify the input of a sampled linear time-invariant system down to one single unit-magnitude input sample.

It is usually fairly simple to build any time domain input from a single unit input (with a linear combination thereof), and thus, given the impulse response for that single input, the output for any time-domain-described input for that system.  Other basis vectors are possible, but may require more or less work (computational effort) for a given type of description of the input.