I can run my sprites 4500+ rpms all day in the summer here in Houston, 110+F out side temp, stock cooling system with no troubles.(less than 200 F If your car overheats you have something wrong.
going from 70 to 60 series tires will not cause your car to over-heat.
I run 175-60 13 Falken ZEIX all-weather tires. they work great..even use them on my spec-sprite racers. cheap too. bout 40 bucks or less if ya shop around.