No, a ceiling fan does not need to be on high speed to cool off a room. In fact, the speed at which a ceiling fan operates does not directly affect the temperature of the room.
Ceiling fans create a cooling effect through the wind chill effect, which helps to evaporate moisture from your skin, making you feel cooler. This effect can make the room feel more comfortable without actually lowering the temperature. Therefore, the speed of the fan is more about personal preference and the level of airflow you desire.
During hotter weather, setting the fan to a higher speed can create a stronger breeze and increase the cooling sensation. However, during milder weather, a lower fan speed may be sufficient to circulate the air and maintain comfort without generating too much airflow.
Remember that a ceiling fan cools people, not the room itself. Therefore, it's a good practice to turn off the fan when you leave the room to conserve energy.