Modeling MOSFET Pole and Transconductance for use in Larger Overall Control System
Is there a way to model any MOSFET Trans conductance amplifier as a single-pole Transfer Function?

In this circuit the FET gate is driven from the opamp's output. If the opamp has the usual single pole compensation, then it has a low frequency pole. Then the opamp's output impedance and gate resistor introduce a second pole together with the FET gate capacitance. This usually makes this circuit unstable, so you need an extra compensation capacitor. If the opamp's output impedance varies with frequency, YMMV.
FET capacitance varies a lot with Vgs, moving the frequency of the second pole around.
FET transconductance varies with Id, which changes the gain.
This is the closed loop response with AC stimulus on input &quot;Vadj&quot;, and stepping DC values on both Vadj and V1 (output voltage).

The closed loop transconductance response has two poles and moves around quite a bit depending on current and output voltage.
EDIT:

What I really want to know is how I can treat the MOSFET as its own &quot;block&quot; in the block diagram shown in my OP. I assume it has the form gm*{something}/(1+sTau)

Not really. Plotting the transconductance of the FET (Iout/Vgs) and its phase:

In a common source, you get 2 poles there. Let's choose a positive polarity for the output current (ie, positive when the FET sinks current), that's easier to visualize.
When gate voltage increases and the FET is in control, output current also increases. More Vgs, more sink current, positive transconductance.
But when frequency is high enough that the FET is no longer in control, because its capacitances dominate the opamp's output drive capability, you can replace the FET with a pair of capacitors, Cgs and Cgd. Some of the gate drive signal passes through Cgd and directly into the output, and this creates a current that is of opposite polarity of what the FET drain current would be.
So, when gate voltage goes up, through Cgd output voltage tries to go UP, which means sink current goes down... So, 180° phase shift, two poles.