Well acctually saturation is not the max number of electrons in the channel.
It`s just the fastest moving.
From gate to bulk there is a capacitance, and as with any capacitor, the higher the voltage on one plate, the higher it is on the other plate, i.e. when Vgs is maximum, electrons in the channel are at maximum and this happens in the linear region.
In saturation, the channel is not completlly formed on the drain side, this is called pinch-off. Pinch-off causes Vd (drift velocitiy) to become larger. The lack of a large number of electrons, also helps speed.
On the other hand in the linear region there is a large number of electrons in the channel and Vd is much smaller:
a. because the channel is fully formed and there is no pinch-off acceleration.
b. The mean time between collisions becomes smaller, because of the large number of electrons in the channel.
The reason it`s called saturation is not because the channel is saturated with electrons, the name is historic. Comes from BJT.
This is also why Id is constant (almost) in the saturation region in respect to Vgs.
Because Vgs is not the main component in controlling electron movment in saturation.
If you want to go deeper than this, I advise reading "Operation and modeling of the MOS transistor" by Yannis Tsividis.
I think it`s more or less the best and most comlete book you can find on the subject.
Everything you ever wanted to know and a whole lot of stuff you didn`t