You're not wrong anywhere. You are one of the few who has a grip on this. I warn you - it's lonely.
This would ruin troops' readiness as fast - faster actually, due to the static nature of the operations - than peacekeeping operations. It would be a waste of active troops.
But I have to say I see nothing wrong with the National Guard, in accordance with it's role as the citizens' militia, helping out. I have suggested several times that all the border states' governors could go a long was towards fixing this problem if they just rotated guardsmen to the border zone. had them throw up a fence on the U.S. side, then guard it.
How about deploying National Guard engineering units to the border along with National Guard MP units to do the police work? The Engineers job would be to begin construction of a WALL. That way we aren't going to have troops along this long stretch of nothing for generations - only the Wall. Also, this construction project would increase the skills of the Engineer units involved, the MP units would get some real training, and the flow of illegals would slow to nothing. Use Airmobile Infantry as backup to the MPs as a reaction force.