Why is the circuit class AC0 unavoidable? Take AC0. 
What is a natural thought process that leads to the definition of AC0? 
Does this class arise intrinsically anywhere? 

My problem is that in the case of unbounded fan-in, AND and OR gates behave very differently. Why do we take these gates along with NOT to define a circuit class?
The class $\mathbf{AC}^0$ arises very naturally as the circuit characterization of problems definable by formulas of first-order logic. A language over the alphabet $\{ 0,1 \}$ is in $\mathbf{AC}^0$ if and only if it is definable in first-order logic in the language of word structures. 
See e.g. Immerman's book for more details. 

$\mathbf{AC}^0$ also arises naturally when studying theories of bounded arithmetic. The simplest of these theories $\mathbf{V}_0$ corresponds precisely to $\mathbf{AC}^0$ in that the functions and relations definable in the theory are exactly those in uniform $\mathbf{AC}^0$. See the book of Cook and Nguyen for more details.