350mA? I think that is the bias of the output transistor. Unlike BJT's, Mosfets in general have a widely varying transconductance (Gm). Gm decreases significantly as the device conducts less Id, say under a few hundred mA. Rod is saying it is important to bias these transistors a 350mA in order to lessen the distortion effects of the drop in Gm around the current crossover. (current and voltage may not crossover at the same time because of speaker reactance)
The gate does not require current like the base of a BJT but rather charge as in like charging a capacitor, and this is dependent on frequency. If you look at a complementary source follower output you see that Cgs is bootstrapped by the output. This means the only voltage change that Cgs sees is the change in Vgs that is related to the change in Id. The real internal capacitance that is the problem is Cgd, but this only comes into play at clipping because otherwise it is much smaller. 15-20mA of current in the driver stage per pair is plenty.