I think you're underselling it a bit. Our ancestral heritage gives us sweating. If that doesn't work, we sort of fall back to fight or flight, with a racing heart rate. move the blood around as much as you can, and hope some part of the body can cool it.

Heat stroke is a real thing, and it's not fun. There seems to be automatic systems you can't think your way out of that make things worse. Fast heartbeat is burning more calories, adding heat, and friction from viscous blood adding more heat. I imagine it's a pretty unpleasant feeling. I tend to go into shock, so I get to dodge most of that horrible sensation.