Electronic – Why a MOSFET in accumulation mode is not conducting

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.

why is it that the required gate-body voltage to create this inversion layer is so much lower than the voltage required to overcome the reverse bias?

Or is it the proximity of the gate (and thus its field) to the area on the semiconductor it is trying to invert?

That's it. The gate is separated by mere 10's of nanometers from the channel by the gate oxide, whereas the channel length might be separated by µm's or more (depending on VDSmax), so the electric field from the gate is hundreds of times stronger.