Does a car run hot from lack of antifreeze?

No, or more accurately very unlikely, most antifreeze would also act to some degree as a coolant which is supposed to make water better at cooling your engine. Having said this the vast majority of cars should run at a good temperature on plain water so long as you keep it topped up and it is at pressure. Overheating would generally point to other problems (valves, sensors, leaks, timing, ect) more significant than lacking additives.

Antifreeze is important to keep the cooling system working, firstly it raises the freezing point of the water to stop it freezing and causing damage in cold weather. It also (again to varying extents) inhibits corrosion and helps keep you system free from blockages. So running without antifreeze can eventually cause your car to overheat but the problem may well not be fixed just by putting some in.