I've replaced just about everything I can think of: defrost timer, evaporator fan assembly, thermostat, and the freezer control switch/dial.

I've cleaned the coils under the fridge and the evaporator coils get cold and ice over/defrost just fine. When I replaced the fan itself it came on briefly but hasn't worked since. The last part I replaced was the freezer control switch which the customer service guy said was probably the last thing he could think of that would prevent it from working; but apparently there's something else left that I'm missing. Are there any other boards or control units that would directly affect the fan or am I just out of luck and need to buy a whole new fridge?

See the attachment for the tech sheet.

I am not sure which thermostat you changed (main temperature control thermostat or the defrost thermostat.

The circuit path for the evaporator fan is
H, red wire
Main control thermostat
Yellow wire
Evaporator fan
Brown wire
Defrost thermostat
Orange wire
Defrost heater
White wire
Neutral

The current for the evaporator fan is low so the defrost heater does not heat up with the fan's current flowing through it.

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.
4. When you start always short the meter leads together. This will tell you that the meter is working and if there is any 0 offset.

There is a good STICKY at the start of this forum about it's use.

Well, this is an odd development: I tested one of the pins on the motor to earth ground and it showed 110V. Touch the GRN/YEL wire to the motor pin and the fan came on. I guess the brown wire isn't grounding somewhere in the path and it's preventing it from closing the circuit. I haven't had a great deal of time to study the wiring diagram, but at least I have something to work with.

Any suggestions from the experts?

28.8ohms according to my meter (not the most accurate one in the world).