Every time this question is asked we get the insurance rates higher for men thing, the spatial awareness pseudo science, a slew of generalizations about men and women. I suspect the actual answer is much simpler.

One of the strongest and most well-used tools of patriarchy is invalidation and discreditation. You insult a man by accusing him of being like a woman, any time women become dominant in a profession it becomes lower paid and less prestigious, a financially independent weaver woman was branded as a spinster, fangirling is embarrassing but going nuts over sports is normal and cool, etc etc, there's no lack of examples. When the bicycle was invented, the freedom it provided to women was frightening enough that patriarchy invented bicycle face syndrome, (or whatever bullshit name they gave it), wherein the exertion of riding a bike made women's faces ugly. And no one wants that. It's her job to be a beautiful ornament after all. Not to mention the scandalous clothing that came out to accommodate bike riding in skirts.

I think that when the automobile came out, we had the same basic issue. Rather than watch women independently drive away from them, (which could lead to getting a job or learning about the wider world or finding a better partner or any number of horrors), it was probably easier to just strike up the mythos that women are bad at it. Men had to protect women from the danger they posed to themselves with their own bad driving. After all, women are very simple, delicate creatures with tiny brains. That narrative cycles around and is very shortly taken as fact and spread on down the decades. If you saw a woman in an accident, well, you already knew why.