I would like to know the historical reason why this is done. But I suspect it's this:
Introducing sampling as multiplication by an impulse train unifies the Laplace and $z$ transforms, and it unifies all four variations of the Fourier transform (Fourier integral, Fourier series, Discrete-time Fourier transform, and Discrete Fourier Transform).
With it, the $z$ transform is just a special case of the Laplace transform; if you know the Laplace transform inside an out, you can derive characteristics of the $z$ transform from the Laplace transform characteristics and from sampling. Without it, they're two independent mathematical entities which happen to share a rather large number of properties.
With it, the Fourier transforms are a family of four. You can start with the Fourier integral and sampling, and you can derive the other three. As with the Laplace and $z$ transforms, you can derive properties of the other three transforms from properties of the Fourier integral, and sampling. Without it, they're four independent mathematical entities which happen to share a large number of properties, and those properties all have to be derived separately.
So even though $\delta (t)$ is a hard thing to get your head wrapped around (and the laxity with which engineers tend to treat it drives mathematicians up the wall), the unity it brings to the various flavors of transform is worth it when you get into the deep math of signal processing.