A very interesting observation. Indeed, people tend to say дождя нет instead of дождь не идёт, in the present tense as well.
The reason is tricky. "Properly", you'd have to use the negative genitive here: дождя не идёт / дождя не шло. But people intuitively shy away from impersonalising a verb like идти which is almost always used personally (i.e. with a subject). So these two conflicting demands — "you need a negative genitive" and "an impersonal не шло sounds rather odd" — find a fairly standard compromise in falling back on a form of не быть instead.
A more recent development of the negative genitive is that its absence (i.e. using the nominative or accusative instead) conveys a sort of definitiveness, much like an article. Thus Вы не видели велосипеда? is "Have you [by any chance] seen a bicycle?", whereas Вы не видели велосипед? is "Have you seen the bicycle?"
In much the same way, modern Russian parses вчера не шёл дождь as talking about "the" rain, in whatever context that might be. (And further context would be required to justify the word order, as opposed to the more natural вчера дождь не шёл, putting the new information last.) "The rain" is clearly a regular occurrence here whose absence yesterday was unusual.
And these grammatical indications are intuitively clear, giving speakers another reason to prefer a simple вчера дождя не было if all they want to say is that it didn't rain yesterday.