There is no magic on holding a set idle speed. It's just a closed loop of slowly closing the idle motor if above 775 RPM and slowly opening the idle motor if below 775 RPM. If the timing of that closed loop has problems then that will cause "idle hunt". It's common that when the lambda system just starts it's closed loop that upsets the idle ECU timing and you get hunting.
The real design problem is when you step on the gas. When you do that the idle motor will continue to close since it sees RPM above 775. Now when you take your foot off the gas the idle motor is closed and RPM falls very quickly and the RPM will drop way below the normal 775 RPM.
The temperature sensor limits how close the idle motor will get to closed. This helps it handle deceleration (taking your foot off the gas). That is also how it can do a cold fast idle but that depends on your curb idle and other engine settings. That keeps the idle motor from closing when you have your foot on the gas. Hence you don't get such a rapid RPM drop and undershooting of the normal idle RPM.
What the idle switch does is quickly open the idle motor more when the idle switch changes from off to on. This is how they are preventing under shoot of RPM when your foot comes off the gas. The ECU still runs closed loop with or without the idle switch on or off.
What this means is you will probably have problems if your temperature sender is not a normal stock unit or is open or shorted. By the way the purpose of the diode in the harness by the idle ECU is just to prevent the 12 volts (and voltage spikes) from the idle switch (opening) from damaging components inside the idle ECU. The grounding of the idle switch signal gets through that diode. They should have put the diode inside the ECU. It also means if your idle switch is not working you will have deceleration undershoot.
I ran into the same problems with my idle ECU design which I'm working on an update now. My ECU gets more problems since the user can select idle RPM and run higher RPM when the AC is on. So I can't fix (make those values a constant) of how closed the idle motor can get.