In places we didn't invade such as Mongolia and Japan, they're actually doing quite well.
Before you look down on the rest of the planet, consider that Europe is one of the best places to survive on earth. It's temperate, it's fertile, it rains a lot, it's geologically stable. Is it surprising that life is easy in Europe?
Compare that to North Africa, which is one of the least hospitable places on the planet: a huge desert. Installing western culture in North Africa would not fix anything.
Many people do in fact enjoy being "poor", as you put it, as long as they have everything they need in life. The western alternative to "poor" for these people is to work crazy hours in a factory. What kind of prosperity is that?
Americans occupied Japan for years, not to mention first destroying her military, and razing a few cities with fire bombs and then nukes. Mongolia's GDP (PPP) per capita is less than $10k. On average, a Mongolian is four times poorer than a French. The same is true for Russia which has immense natural resources, dwarfing those of Europe. Their ability to manipulate natural gas supplies hangs over the EU to this day. Soviets used to control what was once the breadbasket of Europe. Didn't do them much good. Actually, they managed to induce a famine there.
Europe is rich because of industrial revolution and stable governments (more stable than elsewhere anyway). Invading other places was largely made possible by the technological advances rather than causing them. Before, Europe was nearly conquered by Muslisms[1]. Twice[2]. It was regularly raided for slaves (in 16-17th century[3], no ancient history), and starved of precious metals for much of its history.
What kind of prosperity is that?
The kind billions of people living in poverty today would take in heartbeat. They do whenever given a chance. Going as far as risking their lives to cross the Mediterranean or Florida Straits.
[1] https://en.wikipedia.org/wiki/Battle_of_Tours
Actually, the Americans were on the border of _running out of targets_ in WWII Japan because they had already destroyed all the cities.
I heard, the USA has deserts, too. And have you seen Dubai recently? Everything can be fixed with enough oil. Another example: Russia. Nobody conquered them either and it is the biggest country on earth with only 143 million inhabitants, so I bet, they find a way to feed them. Still life expectancy for males is 62 over there.
Westernization might not sound very romantic to people having it, people not having it, might think very different. Ask people from West and East Germany and you will get the idea.
The biggest plus of westernization/capitalism is, that you can not use VW to produce tanks to conquer the world anymore, because they would rather not have their own factories in other countries bombed.