I think the only problem with this is that the Ajah gained influence when they brought stronger Novices to the Tower, and by extension, there are indications that a Sister who found a stronger Novice would gain some sort of influence as well (i.e. scene where Moiraine tells Siuan that she has found a much stronger channeler than Nynaeve - she doesn't mention this as an aside, but as a politically significant addendum that could shield Moiraine from trouble).

And we already know that age and rank resulted in deference, so you couldn't necessarily browbeat any Sister weaker in the Power.

And ultimately, every Aes Sedai was concerned with finding, and bringing the strongest and most intelligent channelers to the Tower.