There is no 'or'. The Constution clearly authorizes controlling the border and citizenship. It does not authorize going 'whole hog on the whole pot'. To entertain any other idea, is to consider the overthrow of our government for something else.
We cannot 'annex' Mexico. We can conquer them (which would be abrogating our constitution), or we can allow Mexican citizens to vote to become a territory of the United States, then allow the states to vote on whether to let them in.
I think we are on the same page. We know what is happening today. We know who is abrogating their repsonsibilities. My suggestions are hints at where policy initiatives today may lead us. Nothing more need be said.