VICTORY.
So knowing how it suppose to work, I pulled the remote wires (that make for bigger task after I had the fan already installed in hard to reach place) and the lid/fan operates just fine with remote switch.
In my "garage pile of junk" I even found old thermostat, but it is furnace thermostat, when fan needs cooling thermostat.
No way I am spending $50 for factory thermostat, but wall switch is all I care about.
With switch on -the lid opens and fan starts.
With switch off -the lid closes.
Thinking about updating it to 21 Century, small relay and WI-FI plug, who cost about $10 will let me operate it via my android from any place in the World. (those plugs also have schedules and timers)
I already have WI-FI thermometer.