Would a system that simply dribbles water over the roof on dry, sunny days reduce air conditioning energy costs?

Remember before the days of air conditioning when your crazy grandfather or dad would spray water on the roof of your house to cool it off? I’m thinking that might not have been so crazy after all. (Is that because I am a grandfather now?)

I did some calculations and figure a water flow rate of only 2.7 gallons per hour (10 liters/hr) spread over the roof and allowed to evaporate would be equivalent to a 2 ton (24,000 BTU/hr) air conditioning system. But, it would only cost less than $0.01 per hour for water. That is 1/70 the cost of electricity.
This is different from the familiar swamp coolers that spray on the people.

Can someone else please check my thinking? I figured the water would be coming in at room temperature, and would be raised to boiling. Then it would evaporate completely. That is a lot of evaporative heat loss that we get for almost free. That loss would be used to reduce the air conditioning load.

Imagine it looks like a garden weep hose along the south side of the roof ridge vent. It would be adjusted automatically to flow at a rate such that the water totally evaporates before reaching the rain gutters.

Has anyone seen a system like this? Ideally it would be used in arid climates. Maybe it exists already.

What am I missing? All criticisms are appreciated.