You neglect to mention that such drastic measures are being contemplated only because the federal government has so far neglected its duties in this area.
The 'close the borders NOW folks' don't really want military authority on the southern border. Most don't even want an immense wall along it. And nobody wants to invade Mexico. We just want the government to do its job and protect the border. We would be perfectly happy to see the Border Patrol empowered and allowed to actually accomplish this task in the legally proscribed manner.
Good point. Long-term having troops on our borders is an inordinately costly and inefficient use of manpower. However, it is only because the crisis of illegal immigration has been allowed to get so out of hand that the military has now become the only realistic stop gap measure before permanent solutions can be phased in.
Another powerful reason to have the military on the borders is to “reduce” the possibility that terrorists will capitalize on our lax border control and slip into the country undetected. It is incredible that our borders remain inadequately protected when so many terrorists from around the world are plotting for our destruction.