Why would we want to annex Mexico?
Well... I could think of one good reason. The biggest reason that the cartels OWN Mexico and access to the border with the U S is that the Mexican military and just about all the elected officials are in bed with the drug lords.
If for no other reason than to clean up the policing of Mexico and the smuggling of people and narcotics.
If we had the authority to tackle the bad guys South of our border it would be entirely possible to clean up that whole mess.
I've been in Mexico a lot and it is corrupt from top to bottom. I believe the citizens of Mexico would gladly have a dependable military that wouldn't rob them and take payoffs from the cartels. If we regulated their elections their society would be fair and much more honest in short order.
I'm not saying that Mexico's problems would be solved over night but just getting rid of most of the corruption and a free hand at access to the cartels would make our Southern states much more safe and there would be no need for a wall at all.
The vast majority of Mexicans living in their country are good people just trying to get by. I have no problem welcoming them into our sphere of influence.
North America would certainly be a lot safer for them and for us.