Sweating works well in dry climates, not as well in humid ones. In high enough humidity, the sweat doesn't evaporate, doesn't cool you, and the whole system becomes a moist, messy loop.
In dry, hot climates sweating works well, but increases our water consumption... these are the habitats where water is often scarce.
Tradeoffs.
I know sweating isn't perfect, for the reasons you mentioned, but do we see nature evolving any better solutions at a mammalian level?
My armchair understanding is sweating is a natural marvel with nothing else like it seen in creatures as large as us. That it truly is a differentiator between humans and other animals (or mammals at least). Sweating is the foundation of humans ability to engage in long-distance endurance running which is one of our unique features, also called persistence hunting[0]. You can easily draw a line between this sweating and our ability to travel large distances and hunt the way we did, which in turn can be connected to humanity's success as a species on earth.
African wild dogs, dingos and other canines can persistent hunt in similar hot conditions. They're much faster than us, so it's not as extreme a strategy as human "track and spook" methods... but arguably better.
Elephants manage to carry those huge, heat retaining bodies. Our Savannah nurseries are full of megafauna like us that live in these conditions effectively.
Some argue sweating, others that we outsourced digestion (fire cooked food), others that we are bipedal, ... not to mention later stuff like language/intelligence.
Also, cats and dogs do sweat a little, mostly through their paws.
I wonder how much of the species' worldwide dominance can be attributed to the evolutionary diceroll that placed our ancestors in subsaharan Africa? Could we have survived in Egypt, Mesopotamia, or India without sweat?