From Lakota America: A New History of Indigenous Power, by Pekka Hämäläinen (The Lamar Series in Western History; Yale U. Press, 2019), Kindle pp. 211-212:
Viewed from Washington, Mexico seemed a natural addition to a nation that was exploding toward the Pacific by way of Louisiana and Texas; the question was the exact size of the appendix. There was a broad consensus that the Nueces strip and San Francisco Bay should be included, and 37° N was quickly established as a new northern boundary. But opinions differed widely on the southern one. Most in the room insisted on a direct line from the mouth of the Rio Grande to the Pacific along 26° N—a thousand-mile-wide chunk of land that would have turned ten Mexican states into U.S. soil. Some argued that they should simply take the whole of Mexico. In the end cooler heads prevailed, and the United States annexed only New Mexico and Alta California. The Treaty of Guadalupe Hidalgo transferred half of Mexico to the United States.
There was a direct mental strand between the Indian Removal Act and the Treaty of Guadalupe Hidalgo, between the debates over the Indian Removal and the debates over the Mexican annexation. Key U.S. policymakers had developed an imperial mindset, a collection of attitudes, convictions, and habits that allowed them to see humans, nations, and entire societies as pawns on an immense geopolitical chessboard. Jackson, Polk, and their officials poured over maps and drew new lines on them, rearranging the world into a new shape. Indian Territory in the West was now the home of removed southern Indians and El Paso was now a U.S. town because they had drawn the lines so. It was in many ways but a reiteration of the centuries-old European imperial ethos which had pushed Spain, Portugal, France, and Britain to carve up the world into empires, but it was new to Americans, and many of them reveled in it. Washington’s dream of an expanding empire of liberty, Jefferson’s utopian schemes for Louisiana, and Monroe’s famous doctrine had been early versions of the imperial mindset, but it was only in the 1840s that Americans possessed the means—sufficient administrative capacity and an ability to borrow money on a vast scale—to start turning imperial abstractions into reality through state-sanctioned violence.
The continental grasslands figured only marginally in these designs. Policymakers in Washington looked straight through the plains nomads into New Mexico and California where the imperial stakes were highest. Seemingly devoid of extractable wealth (except for furs) and securely embedded within U.S. borders, the Great Plains became a kind of halfway house where the Natives could learn the arts of civilization in splendid isolation under the tutelage of a select group of missionaries and government agents. A line of forts running down from Fort Snelling to west-central Louisiana marked a “permanent Indian frontier,” demarcating a sizable reserve in the heart of the continent where the Indians could safely modernize themselves.