I think in this case you're the one with the innate ability to bend the Pattern to your will and there's no "competition" from other strong willed (in the sense of the Power), enlightened people who you need to work against. You, as a channeler, alter their reality by virtue of your innate link to the One Power and the Pattern and in this case the weaves are simply the abstraction you use (like an authoring tool) to understand what you're doing. In the presence of other channelers, the weaves help "convince" them of your version of reality.

The male/female split in this case could be explained by the fact that in RJ's universe (and perhaps our own, too), men and women think very differently, so differently that they need a different abstraction (and don't compete with one another).

When the Dark One allows people access to the True Power, maybe what he's really doing is offering them a bit of his understanding of the Pattern and an interface to *his* Will. This prevents other channelers from seeing what they're doing or interfering with it (at least very well.)

The whole idea of saidin/saidar and the need to channel and form weaves is so heavily ingrained culturally that everyone truly believes in the requirement, which makes it a reality. Sort of how Tower channelers must use a throwing motion to hurl fireballs, but Wise Ones needn't.

I agree it's got some weak spots. I was hoping for criticism. :-)