Somehow I don't think Iturbide's empire is really the key to a stable early Mexico that can do any significant amount of industrialisation. But maybe nothing is, it's more a matter of how less bad things are compared to OTL. When you have Native tribes raiding with impunity to not much further north than Mexico City itself, you have a wreck of an economy, and you have multiple provinces in rebellion plus frequent military coups, that's not conducive for anything but "failed state". That's OTL Mexico in the 1830s/early 1840s. Mexico has a decade to get a better result than that.
Funny thing, the empire (and New Spain beforehand) actually had established relations with the Comanche in a way that was like the Russian empire with its Cossacks. Relations only broke down after the establishment of the republic. And as a side effect, El Norte lost half its people from Comanche raids.
The other problems of revolts and bankruptcy are a runaway side effect of Mexican politics that both set the precedent for revolts, and an unstable political situation at home. Hence the idea would be to avert these crises by keeping a stable regime in charge. The empire seems like a good place to start, as Agustin tried to be the middle ground between the Liberals and Conservatives.
If he can keep it that way until the state institutions are strong enough (tricky but not impossible I think) then you can get a much stronger and bigger Mexico that can project power. How much stronger? Well that depends on internal politics.
And I'm not sure why Cuba might want to join Mexico when they're fighting a war against Spain for their independence. Unless I'm wrong and there was a pro-Mexican movement of any note in Cuba at that time? And if Iturbide's in charge, wouldn't that mean the conservatives have the upper hand in things in Mexico? And weren't many of the Cuban independence fighters liberals? And Mexico of course won't find any support from the pro-Spanish portion of Cubans.
It depends on how internal politics in Mexico turn out. I'm assuming the Mexicans dont aim for Cuba for a few decades onward, which means the state won't be the same. I think it will lean conservative because of the economic situation (large landowners, low class mobility) but that's not a given.
Cuban leaders were usually liberal, but well they might be willing to work with a conservative government to bolster their agenda. Pragmatism makes for strange alliances.
It's all a gamble, and one that depends on certain things going right for both parties but I don't think Cuban leaders flipping to Mexico is out of the real of possibility
Yeah, I totally forgot the whole slavery issue in Mexico when I typed my previous post. Trying to abolish slavery in Cuba seems like a hassle that'll just make Cuba try to split off like Rio Grande or Yucatan since the political elite won't go for it without compensation of some nature.
I don't think the elites are a problem- if Mexico intervenes during a revolt, rather than a direct conquest. That way you can replace the elites with your own proxies (the liberals/independence faction) and purge the unwilling ones.
Although compensation for slavery has been done before in Colombia and Brazil, so its not out of the realm of possiblity.
Invading with the help of a slave uprising would look horrible--and what's to stop some slave from trying to make his own version of Haiti there?
Why bite the hand that's feeding you? The slaves lose too much if they try to fight back against Mexicans that are helping them.
They do have to get past the Spanish Navy though, and that includes supplying their invasion of Cuba. But it would be good for Mexico to make an effort against those Cuban bases.
The South might not want it as much as they did, but the proximity to the US would be tempting. If both Yucatan and Dominican Republic attempted to join the US, then I thing Cuba would definitely have a movement. And it's so close, after all, and it's just better land than either Yucatan or the Dominican Republic. All those free blacks could make a good cheap labour force that would be among the cheapest in the country (but not as efficient as slavery). And lawmakers who might be concerned of what happens if slavery is abolished in the US could do a "testbed" of sorts for Jim Crow-style laws to see if they accomplish making a good social order. The economics would also mean Cuba would be very tied to the US at some point. I think if even if Mexico can keep it, it'll still have very strong ties to the US.
DRs attempt is a bit of a fluke given only Baez and his cronies wanted it, and they were a minorty of landholders in the Southwest. Its not representative of a general Caribbean trend to petition for annexation.