The problem is actually high heat combined with high humidity. Sweating fails to cool you down and you cook. Broiling is very high heat and low humidity.
That would be "grilling" in most parts of the world
No way. Grilling implies the heat comes from below while the food sits on a grill, as opposed to broiling where the heat comes from above. Neither is a perfect description of humans cooking due to global warming, but our heat definitely comes from above.