The temperature sensor will have some dead time. A simple turn on and off with a narrow temperature band will not work well. The rate of change in the temperature will vary depending on the level.
One more reason that you may see the OL contact in the neutral circuit is when you have reversing starters with one OL relay. You can't always break both coil circuits on the "hot" side with one contact, but you can interrupt them both on the neutral side.