Why do we use a wide-band amplification in order to reproduce a sharp pulse without distortion?
As the wikipedia page says it:

Generally speaking, the more concentrated f (x) is, the more spread out its Fourier transform f̂ (ξ) must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in x, its Fourier transform stretches out in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform.
The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.

So having a pulse with very sharp transitions means it will have a very large spectrum, thus wasting precious bandwidth (and higher EMI). So, in order to limit the bandwidth of the pulse, while still having a pulse, you simply cut off the higher harmonics, or (lowpass) filter it. Whether it's a raised cosine, Gaussian, or whatever other filter, that's up to the special requirement of the task needed to be performed.