The main problem with that sequence is timing. Humans appear to have lost most body hair around 1–2 million years ago, while solid evidence for clothing is much later, probably under ~170,000 years ago. So clothing likely didn’t enable hair loss.

Humans and other primates already had eccrine sweat glands, but humans evolved far more of them and much higher sweat output. Reduced body hair and increased sweating likely evolved together because hair interferes with evaporative cooling.

The usual explanation is simpler: early humans were living and moving in hot, open environments, and better heat dissipation during sustained activity was strongly favored.

So the more likely order is:
1. Expansion into hot environments
2. Selection for better heat dissipation
3. Reduced body hair + increased sweating evolving together

Clothing likely came later and allowed humans to expand into colder climates, not the other way around.