Scientists are questioning whether wild horses populations in the Americas went extinct and some Native Americans will tell you they didnt. Native Americans did not buy and breed.

https://www.yesmagazine.org/environment/2020/04/27/native-horses-indigenous-history

There was no mass release of horses at the end of the Civil War. Horses died by the millions in that war and at the end they needed to obtain more horses.

https://en.wikipedia.org/wiki/United_States_Army_Remount_Service

The US Cavalry still existed (and does exist) and still had to function (American Indian Wars)

The mustang is a free-roaming horse of the Western United States, descended from horses brought to the Americas by the Spanish.