Bingo! IMO the whole basic problem most of Latin America faces is the who peon and patron thing they inherited from the Spanish. I mean other than Chile and Costa Rica what other former Spanish Colony in the Americas is not a basket case
According to the UN's Human Development Index Mexico is actually at the tail end of High Human Development and ahead of countries like Russia and China. Mexico's not far from what we want. Keep the PRI out of power for another few decades and it just might make it, too. Mexico will likely never be a major military power. Culturally it's just not in the cards; with the US as an ally and no major foreign threats they really don't need to be anyway. At most, Mexico might participate in a few UN peacekeeping missions and make Guatemala think twice about its opinion of the border between the two. But as an economic power Mexico is already fairly formidable. Reducing corruption and infrastructure development are the order of the day today, as they were back in the days we're talking about. Mexico didn't even have a highway system until the 20th century.
Keeping Iturbide on the throne would work, but he really needs a strong and respected personality to fill a prime minister role to keep him from going too overboard against dissenters. His overly harsh responses to people who disagreed with him is what got him run out of the country (and shot when he tried to return) in OTL. Make your POD a little earlier and have Father Hidalgo survive; he'd be a good choice for Iturbide's PM (and would keep the Church on the side of Mexico). Being very popular with the Indian and mestizo populations, Hidalgo's participation in the government would allow the Iturbide dynasty to become a popular institution with the lower classes. Hidalgo was also a social reformer, which might help mitigate the near-caste system set up under Spanish rule. Mexico had already abolished slavery as of 1821 so that's a good start. From there, they just need to make the peon/patron system less like serfdom and Hidalgo's just the man for that.
A Hapsburg regime wouldn't be out of the question, actually: after independence the Mexicans petitioned every Catholic royal family in Europe to take the throne, but none would accept (which is why Iturbide ended up on the throne in the first place). The Hapsburgs are Catholic and were at one point the ruling dynasty in Spain. A restless Spanish Hapsburg collateral who needs to be put far, far away would be an ideal fit for the Mexican throne. It wouldn't be the first time that the Spanish Hapsburgs dumped a troublesome nobleman on Mexico, after all.
As for Latin American countries being well-off: Venezuela isn't all that bad off, their present leadership notwithstanding. And Argentina has been close to major power, if not Great Power status as recently as the early 20th century (the Peronistas didn't exactly do them any favors after Eva died). With civilian rule restored and a couple more decades to get their economy in order Argentina could be a strong nation again.