From the thread title I thought this would be a question about plate teutonics. Boy was I wrong!
Got a bit of an axe to grid, have we? This is GQ, not GD, so I’ll try to answer your question as such.
First, let’s clear up some misconceptions and give you some much needed historical background. When the Spanish, Portuguese, French, English, and to a lesser extent the Dutch and the Russians started colonizing the Americas, they all had to deal with the Natives in some way or another. In Central and South America, there were large, advanced civilizations (Aztecs and Incas, respectively, plus numerous other smaller groups). The Spanish basically conquered and enslaved these civilizations. In what is now the US and Canada, no such advanced civilizations existed, just scattered, nomadic tribes. As more and more European settlers (primarily English and French) arrived, those tribes kept getting pushed further and further west. At the same time, the Spanish were extending to the North from their power base in Central Mexico to control California, Texas and New Mexico.
Flash forward about 200 years. The US, now independant of England is still pushing westward, filling in the land acquired in the Louisiana Purchase. Also, many pioneers from the US are moving into Texas and California (now belonging to Mexico, also newly independent from Spain) seeking desirable land. By this point the Natives were not much of an issue, pretty much relegated to living on reservations, with little choice but to accept whatever the US government was willing to give them. You can argue that they got a raw deal, but this forum is for factual answers, not debates.
Okay, now for the answer to your question: Why didn’t the US take over all of Mexico? Well, it wasn’t for lack of trying. Remember the settlers from the US who were moving into Texas? Eventually, they started outnumbering the “locals” (who were in turn descendent from the earlier Spanish settlers). Dissatisfied with being under Mexican rule for various reasons, Texas broke away from Mexico in the 1830s. The US played no part in this war. Eventually a watchful peace (to borrow a phrase from Tolkien) emerged, with Mexico sort-of-but-not-really acknowledging the independence of Texas. Texas petitioned the US for annexation, but not wanting to risk war with Mexico, the US declined. Texas instead declared itself an independent republic. Mexico never formally recognized their independence, and the area between the Rio Grand and Sabine Rivers was disputed territory.
During the US presidential election of 1844, James Polk promised to annex Texas, settle the Oregon question (the territory was disputed with Great Britian), and to purchase California from Mexico. Polk won the election by a landslide. Seeing the writing on the wall, the outgoing president, John Tyler, signed the bill to annex Texas before leaving office. US troops under the command of Zachery Taylor were sent to fortify the Texas border, specifically the disputed area between the Rio Grande and Sabine Rivers. Mexican forces attacked those troops, and the US-Mexican war began. It lasted from 1846-1848. US settlers in northern California took the opportunity to declare their own independence from Mexico, but that only lasted about a month before US forces showed up and placed the area under military jurisdiction. Eventually, US forces occupied all of California and New Mexico territories.
Eventually US Marines landed near Vera Cruz, pushed their way inland, and captured Mexico City. Polk sent diplomats to negotiate a peace treaty. Due to the disarrayed state of the Mexican government, there was no one to negotiate with them. Polk eventually recalled the diplomats, and intended to continue the war. However, in the intervening time between Polk issuing the recall order and that order arriving in Mexico City, the Mexican government got their act together and negotiated a peace treaty that ceeded all of Texas, California, and New Mexico in exchange for $15 million. Polk had no choice but to submit the treaty to the US Senate, who ratified it, ending the war and nearly doubling the size of the US.
So the answer to the question is they tried, and nearly succeeded.