How effective were the Spanish in gaining control of Latin America?
I would have to say that the Spanish were very effective in gaining control of Latin America. After all, they controlled pretty much everything from the Valley of Mexico to Patagonia (not including Brazil) for over 200 years.
Of course, the Spanish were not able to gain complete control. They did not have enough Spaniards to control all of the rural areas and many natives lived more or less outside of Spanish control even up to the time that the various countries became independent. But in terms of the heartlands of the various countries, the Spanish held firm control and were only really challenged for control by other people who were essentially Spanish (the creoles).