Did they buy it? No. They took it from Mexico as a consequence of the war. In the case of Nevada and a lot of the other western states, government ownership of the land predates the creation of the state.
This is my problem. If "I" the government own land and then "I" allow brother Nevada to draw boarders and put in place articles of corporation and then become a member of the "United States" which includes the protections of the Constitution how do "I" now have claim of any land within those boarders?
Hmmm, I don't see your point other than the western states are still getting screwed. Past precedent calls for remedy.
How many states were states before the Louisiana purchase in this map?
.
Then look at Federal control of that land.
Actually as part of the treaty to end the war they paid Mexico $15 Million.