Comparison between MOSFET, MODFET, and MESFET?
I can only answer the first of your two main questions.


  Why would you want to use a MOSFET when the MODFET supposedly has a higher carrier mobility?


Cost. First, if you want to make a GaAs/AlGaAs MODFET, that means you're working with GaAs wafers rather than silicon, and these are much more costly per device. Even if you make a MODFET with varying Ge content in SiGe, you are working with a more exotic and somewhat more costly material system than ordinary silicon. 

Secondly, because the modified composition layer (AlGaAs or high-Ge SiGe) must be added by epitaxial growth, which is a slower and therefore more costly process than the ones needed to form an ordinary MOSFET on silicon.

Thirdly, because vast amounts of research have gone into reducing the cost of silicon CMOS processes over the past decades, allowing CMOS to be produced in high volumes, with high yields, and thus lower costs.