For simplicity, in this thread, let's call "saturation" == voltage saturation, the low voltage, low incremental resistance region on the output (Vce vs. Ic or Vds vs. Id) curve.
(For historical note: triodes have a fairly constant output resistance, so by this definition, would always be in saturation, even in the "linear" range (which they're never really out of, either*). This is interesting, because triodes appear to be the only "constant voltage" amplifying device -- all others are either constant current (three or more terminals: pentodes, transistors) or negative resistance (two terminals).)
What characteristics are desirable in switching? The output voltage should be either a good solid '0' or a good solid '1'. You can get a good solid '1' in cutoff, because the voltage simply rises to its open circuit value, regardless of resistance.
But you can't get a good solid '0' in the linear range, because the voltage is practically undefined -- supplying only a constant current, if the load resistance varies, the output voltage varies proportionally. A pretty crappy zero! However if we saturate the device, it can sink extra current, accounting for a possible change in load resistance, and maintaining a good solid '0'.
It's worse. The output current of a BJT or MOSFET is poorly defined, being dependent on hFE and Vbe, or Vgs(th) and g_fs. Both of which vary dramatically between component size, manufacture, and temperature! How can you possibly address all of these variables during the design phase? By saturating the device, so that all of them get squashed away!
Tim