The US population expanded from the North East where the Industrial Revolution began. Economic jobs boomed and so did wealth. California created a tech boom, and Texas created an oil boom. While southern and middle states lack technology, industry, precious metals.
Major European countries (cold) joined forces economically to prevent ruin, while middle eastern countries (warm) face USSR style corruption.
In Russia, most of the countries resources are reverted to Moscow or St Petersburg. While warmer AND colder areas of the country lack basic funding/upkeep.
South Africa has the highest GDP in Africa primarily due to historical colonialism stabilizing the area and turning it into a major port. While surrounding countries are 1/10th the GDP due to govt policies and corruption.
The east side of Australia is heavily built up compared to the rest of the country. This is due to port access and farmland while the west coast is much more dry and rocky.
Obviously this isn’t exhaustive, but serves the point. There is no single answer and “warm is poor” only applies to VERY select regions.
As a person who grew up in a hot tropical country and is now living in a cool temperate one, I found it much easier to do things in a cooler climate than in a warmer one.
This is anecdotal of course. Take it for what it is, just my personal experience.
Warm climates may have more resources, but the wealth concentration is somewhat artificial; nor that cold climates had difficulties in centralizing power.
Instead, the simpler and more accurate narrative is that cold climate countries are more resource starved, therefore have more incentive to plunder the resource-rich nations. The easiest nations to plunder are the technically lagging nations.
And what lead to a technology-savvy culture in the recent past? War, with the occasional dose of peace.
Egypt, Mesopotamia, the Indo-Gangetic plains, and the Yangtze are all in decidedly warmer climes than much of Europe, and the civilisations they birthed still endure.
In cold climates you can do fuck all during winter, and frankly, most of the year. Cash crops don't grow at all. All the water is frozen solid for months on end. There is hardly any sunlight. This means no plants. No plants means no animals. No plants and animals means no food. There's a reason why Antarctica, Greenland, Alaska, much of Canada, Siberia, the Tibetan plateau, and Patagonia remain completely undeveloped.
Many tropical and sub-tropical countries were extremely wealthy until at least the 1600s. India was amongst the richest countries in the world until the British colonised it, squeezed every last drop out of it, and then dropped it like a hot potato once Britain itself was nearly bankrupted and its infrastructure in smithereens following WW2 (and sliced it in half for good measure, setting the stage for an infinite casus belli between the Islamic Pakistan and the secular India). Belgium did something similar in the Congo, and there's probably a cathedral full of Congolese hands and legs somewhere in Brussels.
The early American colonists arrived to see most of the native Americans ravaged by smallpox and the plague, foreign diseases the latter had no immunity to. They saw open land for the taking, and called it manifest destiny in their race from Virginia to California and Oregon. They sequestered the remaining native Americans (who to this date are misnamed as 'Indians', another facet to the enduring racism and prejudice in the US) into 'reservations' and 'nations'. Likewise for Canada.
In central and south America the violently evangelical Catholic Spaniards and Portuguese took advantage of the fractious rivalries, and picked them off one-by-one usually allying with one empire against another, rinse and repeat—from the Aztecs, Maya, and Toltec to the Inca.
The large majority of modern European countries have built their current wealth on the backs (and blood) of exploited and colonised peoples from Central and South America, to almost all of Africa and the Middle East, to South Asia, Southeast Asia, and East Asia.
In a just world, Spain, Portugal, the UK, France, the US, and the Benelux countries would collectively be forced to pay reparations amounting to about five to ten trillion euros/pounds/dollars per annum to their ex-colonies, slaves, and exploited peoples until the latter returned to a semblance of development with everyone above the poverty line, and with reliable electricity, sanitation, food supply, housing, and Internet access.
In that past India, China, Mali were massive economies and those didn't happen overnight. They took millenia and the weather stayed warm throughout that time.
As always it is often technology which creates wealth and whoever has the "current" tech usually shoots up for a few centuries and then the cycle repeats itself.