The problem isn't the form of government that Mexico had. Mexico has immense challenges to overcome. Any kind of system of government is going to run into the same problems.
1) Mexico's population is vastly illiterate and uneducated. This hurts both political and economic development.
2) The elites are bitterly divided over the role of religion and the Catholic Church (including its land holdings).
3) Mexico is a huge country that requires much power to be devolved to the local elites, but any central government wants to centralize power in its own hands. This creates a huge tension within the country and is a major source of conflict.
4) Mexico's geography lacks the long navigable rivers and lakes that enable water transportation and canal building to speed up economic development. Before the rise of railroads, Mexico has little opportunity to develop its economy nationally.
5) Most of the land is held by informal communal agreement (either by traditional among the villagers, or in coordination with the local Catholic Church lands), but lacks formal title. It is a feudal economy on the cusp of the modern age. Moving towards any system of formalized title will displace much of the agricultural workforce and end any kind of economic independence of the peasants.
6) Mexico lacks the native capital to fund its internal improvements, so it becomes reliant on foreigners for investment which creates a lot of political problems familiar to anyone who knows the dangers of absentee landlordism and nationalist backlash.
With hindsight and a level of control of computer games, someone could devise a policy that can make Mexico into a decent, regional power (but not a world great power). However, no such hindsight exists, and such control is impossible in real life. Even if we create a new person with the qualities we want or handwave away personality changes of an existing person (like Santa Anna, Juarez, or Diaz), that person still needs to deal with immense political opposition. We might see 6-12 years of wise, benevolent leadership, but continually for decades?
It's important to note the many wars Mexico had during this time, most of which were due to the issues above.
War of Independence (1810-1821)
Central American Revolt against the Mexican Empire (1822-1823)
Attempted Spanish Reconquest (1821-1829)
Various Rebellions against Santa Anna (1835-1848)
Pastry War (1838-1839)
US-Mexican War (1846-1848)
Reform War (1857-1861)
French Invasion (1861-1867)
Several Rebellions under Juarez, Lerdo and Diaz (1871-1876)
Mexican Revolution (1910-1921)
Cristero War (1926-1929)
There is a reason General Sherman argued against the Confederate rebellion by saying it would lead to the same fate as Mexico (endless war). The above list is indicative of the huge political challenges in ruling Mexico and coming up with a settlement that can satisfy everyone. Between 1810-1876, there are only a few years when Mexico wasn't engaged in some kind of internal war. Porforio Diaz created a peace for about 30-40 years, but it exploded again at the end of his term because he hadn't really settled any of the issues that had created political instability beforehand. Even today after decades of stability with the PRI and some serious political and economic reforms fromt he mid 1980s to today, Mexico has achieved a level of stability and prosperity, but we still see the same problems.
Mexico simply can't become a great power by 1890.