It is a combination of a few factors.
I doubt they turn off the heat on their deep fryers when not in use
Some do it. There are small deep fryers which have the capacity for one portion of fries, and bistros and restaurants where you sit down and wait for your order to be prepared turn them off during lull times. They take some minutes to heat up, but the customers are expecting the wait anyway. Of course, fast food joints can't do this.
Is it the specialized equipment they use
Partly, yes. Your stove is engineered to output a constant amount of energy (time modulated), no matter what you put on it. Different cooking vessels with different types and amounts of food heat up at a different rate and reach a different equilibrium temperature. While a commercial deep fryer will allow you control the temperature too to some extent, the fryer has a known shape, material and volume, allowing the engineers to work with a narrower possible range of temperatures. It will only overheat in some less usual circumstances, such as letting the oil go very low while being turned up very high, but the typical setting and volume are likely to be chosen in such a way that the final temperature is optimal for deep frying and too low for overheating.
It is likely that you can reach this temperature with your stove too. From your description, it sounds like you choose a stove setting which is too high, and the oil doesn't reach an equilibrium temperature, but continues to heat up while you're frying. Also, it seems that you are using relatively little oil compared to the mass of the food, so that the thermal mass of the food cools it down and it doesn't burn while frying. Check the oil temperature with a thermometer and adjust, and you will probably find a stove setting which can hold 1-2 liters of oil at a relatively stable temperature after a sufficiently long period of preheating.
Or is there more to it?
The other part would be the fat itself. If you are reaching for your standard high linoleic oil every time you want to deep fry, it can be that its smoke point is lower than the optimal frying temperature. These places use fats specifically designed for deep frying, with rather high smoke points. So even if their temperature fluctuates between, say, 160 and 190 Celsius, it still doesn't smoke.