a mile is 1760 yards. very few can make such a shot reliably and repeatedly
there’s no reason not to schedule 10+ troops to monitor/guard each mile in 3 shifts (3-4 active per shift), overseen by a command structure that allows for backup to be deployed where needed.
A fence is a good solution for Arizona, New Mexico and California, where the border runs cross-country.
Texas and the Rio Grande River, however, present a separate problem. And it is instructive how Israel has defended the Jordan River border between the West Bank and Jordan. The desert terrain is quite similar to Texas, as well.
Israel built a border fence that resembles a stock fence, about 6' tall with gates, about a kilometer back from the river. It's not intended to be an obstacle but instead serves to define the border area. The actual border defense is provided by frequent observer posts and inter-linked strong points with machine gun positions. Only a small percentage of these are manned at any one time, but reinforcements can be emplaced rapidly.
Sounds pretty much like what you're suggesting.
My eyes are no longer what they once were but I am confident I could keep you from advancing upon my position. The misses would be close enough to make you wonder where the next one will hit.
Anyone who takes the time to view those 85 videos and spend a few days at a 500 yard range will be able to keep attackers at bay unless they are also well prepared..