Some confusion about impulse response
The answer is in your question: we're speaking of Linear Time-Invariant systems. Their properties are linearity and time invariance.

Linearity

$$\text{if } x_1(t) \text{ produces } y_1(t) \text{ and } x_2(t) \text{ produces } y_2(t)\\
\textbf{then}\\
\alpha\cdot x_1(t) + \beta\cdot x_2(t) \text{ produces } \alpha\cdot y_1(t) + \beta\cdot y_2(t)\\
\forall\ \alpha ,\ \beta\in\mathbb{R}$$

Time invariance

$$\text{if } x(t) \text{ produces } y(t)\\
\textbf{then}\\
x(t + \tau)\text{ produces } y(t + \tau)\\
\forall\ \tau\in\mathbb{R}$$

So if you know the impulse response, and call it \$h(t)\rightleftharpoons H(f)\$, then it is clear that for an arbitrary input, if you can express it as a sum of delayed impulses, the output will be a sum of delayed impulses responses.