@flo Hot tap water, if set at the recommended water heater temp, is only 120 degrees. (In my last post, I didn’t mean that you were to use hot tap water. I only meant that regular water, not salted or sweetened or combined with any other chemicals or substances, and not in your car’s radiator or in a pressure cooker, otherwise boiling point would be higher than 212 degrees.) I refer to boiling pure water @ 212 degrees at sea level.
And you’re only rapidly cooling the eggs before storing in the fridge so the bacteriological problems you’re concerned about aren’t an issue. (You’re not storing the boiled eggs for any extended period of time in the bucket.)
Use a thermometer to see how cold your tap water is. Mine is 49 degrees. (Oh, yes, that is degrees Fahrenheit.)
If the bucketed water turning warmer than room temp by too much, use more water in the bucket, like, say, 3 gallons. As soon as eggs are cool, take them out of the bucket, dry them, and store them in the fridge.