Basic MOSFET modelling question
What you're talking about is an electrical model of a transistor, not necessarily a fundamental equation of how semiconductors work. As far as models go, it's a fairly simple one, not taking into account many behaviors and parameters. Typical models for computer-based simulation, such as BSIM3/BSIM4, have hundreds of parameters to more accurately model what the transistor is doing under more conditions. 

Your example equation - 

$$ I_d = \dfrac{1}{2}k'_n \dfrac{W}{L}V_{OV}^2$$

is a simple model useful for evaluating analog designs. Its use is in understanding relationships and dependencies for various parameters, such as bias currents and transistor sizing. 

There will be some behavior that your device will have that is poorly modeled by these equations. Something as simple as non-constant output resistance in saturation might be an example, or maybe it does something really funky. There may be odd temperature dependencies that are unexpected.

When doing some "not silicon" analog design, I've seen that these simple models are sometimes inadequate. The basic premises that they have are still useful.