EMALAHLENI, South Africa—The gleaming new chimneys and hulking generators of what will soon be the second-largest coal-fired power plant in the Southern Hemisphere are at the center of South Africa’s $20 billion, 50-year bet to fuel economic growth and lift its population out of poverty.

Now, South Africa and its Kusile Power Station have also become a test case for whether rich nations can help poorer ones transition away from the dirtiest fossil fuel.