Mosfet channel length modulation
The channel isn't completely cut off due to the strength of the electric field between the drain and the channel. We use this region because the current is no longer dependent on Vds and we can amplify waveforms without distortion as long as we make care to stay in saturation.

I mean strictly speaking you're right the depletion region around the drain in saturation is smaller than it is in the triode region, so it technically isn't as good, but as for the reason stated above it doesn't matter, current is still flowing, but now largely without regard to Vds.