Why did Mexico encourage the settlement of Texas by Americans?

Terrific research on this topic has been done. You can find it here:

http://americanaction.us/index.php/american-history/mexican-american-war/origins-of-the-texas-revolution-of-1835/

Mexico probably encouraged Anglo settlement in Texas for the same reason that Spain had done so earlier: they couldn't get anyone else to go.

Weber says in "The Spanish Frontier in North America" (Yale, 1992):

Eight years after the king issued his order, the first and last contingent of government-sponsored immigrants from the Canary Islands reached Texas.... [which] languished as one of the least populated provinces on the northern frontier of New Spain.

As in Florida, high percentages of Hispanics in Texas were soldiers or civilians who depended upon the military for their subsistence. The military, in turn, depended heavily on the Crown, which paid dearly for its failure to develop Texas economically, as the king's own auditor suggested.

Efforts to encourage Canary Islanders and sympathetic Acadians to settle in Louisiana had achieved results, but Spain could not afford the subsidies that it needed to draw large numbers of Spaniards to its frontiers. When José de Galvez authorized 50,000 pesos to promote immigration from Spain to East Florida in 1786, a desperate Governor Zéspedes spent it to feed his troops. Out of the experiment only five families of Canary Islanders came to St. Augustine.

Unable to attract colonists from Spain or its American colonies, Spanish officials began in the mid-1780s to allow immigrants from the United States to settle in Louisiana and the Floridas and to obtain generous tracts of free land and access to the Mississippi. By 1788 local practice had become Crown policy as a result of lobbying by Gov. Esteban Miró....

The new Spanish policy required immigrants from the United States to take an oath of allegiance, but in a unique reversal of previous policy and practice, it did not insist that Protestants convert to Catholicism.

Spanish officials had by no means been unanimous in supporting this risky and paradoxical strategy of importing aliens and heretics to ward off aliens and heretics.... In a private letter to President George Washington, Secretary of State Thomas Jefferson had praised the liberal Spanish immigration policy as "the means of delivering to us peaceably, what may otherwise cost us a war".

After Mexican independence much of northern Mexico, infact most of many parts of Mexico, were devastated and sparsly populated. These northern areas were hard to defend against both indian incursions and anglo land poaching. Mexico had a just fear that the US was eyeing their northern states. Mexico lacked the people, resources, and equipment to settle these northern areas, so they decided they should try to encourage first foreign (non-us settlements) but when they failed they tried Anglo (US) settlements but to ensure loyalty they would make these Anglos catholic and swear to obey the laws of Mexico. Obviously this backfired as the anglo's loyalty remained with their former home rather than Mexico.

Economically, the country faced devastation in 1821. It stood in marked contrast to the rich colony that had promised great potential towards the end of the colonial era. Money barely circulated. Once-rich mines struggled to regain their former efficiency. Ranches and farms were no longer productive...An equally urgent concern for the young country was guarding its far northern possessions from United States expansion; Texas was especially vulnerable to encroachment from that country, and colonization offered the best deterrent. But Mexico lacked the strength of population numbers to settle the north.

https://tshaonline.org/handbook/online/articles/npm01

Also Indian raids were a very serious concern and vast swaths of Texas were unsafe for habitation because of the raids. If you look at maps of the time, Indians controlled almost the entire Western half of Texas. Mexico hoped settlements would help deter the raids. Where as something like 40-50% of the Indians in California became Christian, this was not the case in Texas.

After the 1819 Adams-Onís Treaty defined the U.S.-Mexico boundary, Spain began actively encouraging Americans to settle their northern province. Texas was sparsely settled, and the few Mexican farmers and ranchers who lived there were under constant threat of attack by hostile Indian tribes, especially the Comanche, who supplemented their hunting with raids in pursuit of horses and cattle.

To increase the non-Indian population in Texas and provide a buffer zone between its hostile tribes and the rest of Mexico, Spain began to recruit empresarios.

http://cnx.org/contents/_8trGL3L@4/Independence-for-Texas

Also remember Texas, especially south Texas and the Llano Estacado are not very hospitable places and so encouraging Mexican immigration there was difficult. Especially when there were better spots that were open because of the depopulation caused by the civil wars.

When Mexico won its independence in 1821, it covered an area from Central America northward to California, Utah, and Texas. In fact, Mexico in 1835 was just about as large as the United States at the time.

To help develop parts of this vast territory, the Mexican government invited settlers from the United States to take up lands in Texas. Mexican leaders hoped these settlers would eventually become Mexican citizens and be converted to the Roman Catholic faith, the main religion of Mexico. These citizens of Mexican Texas could then be expected to defend their land from possible invasion by the United States._

This was the main reason for Mexican encouragement of American settlement in Texas; counter-intuitive perhaps, but was the theory then.