Why take separative quotient in forcing?

Question

In many expositions of topics on forcing, the poset is assumed to be separative, otherwise we pass to the separative quotient. For example, this is the convention taken in this introduction to forcing by Moore, Handbook chapter by Abraham, and the one by Cummings. But I don't really understand the reason behind that. There are certainly some results such as equivalence between Baire and not adding $\omega$-sequences that only hold for separative posets, and also $p\Vdash q\in\dot{G}$ is equivalent to $p\leq q$, but are these essential? In particular, are there any serious issues if we don't take separative quotient at each step when defining iterated forcing?