Of course he did. And so did Eve.(*)
That is, I mean, if you define “lead” as deliberately setting a positive example for one’s companions. On the other hand, if “lead” means to “dominate” or “seize control over”, then... well, actually in that case I must absolutely note that he did fail at that also. Of course, in the first sense, Adam’s failure was regrettable. In the second, highly laudable... yes, despite the results.
Now, what is my point? Why bother with this today, really?
Because it’s a difficult thing to successfully influence others without taking control.
However, to the underlying question about Genesis 3, why did Adam’s sin cause the fall? Was the MAN more responsible than the woman, for their mutual sin? Historically speaking, there probably was that patriarchial assumption among the original storytellers and story-hearers. But in all honesty, there’s probably also a communal aspect of ancient life that's also assumed in the story - one that we individualistic westerners (internet readers, especially) may genuinely be missing.
One naturally wonders, if Adam had eaten first, would Eve have immediately fallen? (IDK, but...) From a storytelling standpoint, having the woman eat first satisfies BOTH the communal AND patriarchal expectations of the ancient world. Objectively, therefore, we might honestly speculate that the timing [of sin's consequence having arrived] wasn't entirely because Adam was male. Even considering the ancient Jewish mindset, the other aspect might deserve equal consideration - that Adam was the other half of their social unit, and when Adam ate forbidden fruit, sin had total sway over their entire group, corporately. Not to speculate too hard, but it's good food for thought.
At any rate, none of that is my point, today. This is: It’s a difficult thing to successfully influence others without taking control. It may sometimes get easier in situations where mutual influence is the custom... but even then, even these days, results are still known to vary.
Dear Lord, open our eyes to what your kind of "leadership" truly is.
(*) Caveat 1: this post inspired by the blogosphere's thorough trouncing of some complimentarian blowhard, about a week ago, a trouncing possibly more set against his fundamentalism than his patriarchalism.
(*) Caveat 2: see my view on Genesis AS IF History.