The analysis is more complicated when you introduce terms such as velocity saturation and channel length modulation. The reason why they would use the ideal square law is to arrive at a conclusion that make sense...it's a lot easier to deduce something from a simpler equation. For example using the basic equation it's very easy to see that overdrive voltage has an impact on the ICMR of a differential amplifier.
By the way, even with introducing velocity saturation and channel length modulation, the square law equation is fundamentally flawed as it grossly underestimates depletion charge(due to linearizing the depletion charge and assuming a slope of unity), meaning that it overestimates inversion charge and by that it overestimates drain current.