I don't think, based on my own family history, that the vague stereotypes you're talking about had anything to do with why women did or didn't become programmers early on. Do you assume men weren't involved in the textile industry?

What happened (anecdotally) is that early in the history of computers, it wasn't possible to get into an engineering program for any reasonably bright woman. They're like "of course we admit women, but we already have a woman". The punch line being that they didn't even have one woman as an undergraduate. So maybe you go to a woman's college, and maybe you study math, and then you end up working as an assistant to an engineer, who is analogous to a secretary, taking engineering problems and coding them.

[Edit: I have or had a picture of the person I'm describing with a non-industrial knitting machine, but probably considerably later than the 50s when she started programming]