What you describe is basically what we're doing now, to the extent we're willing to fund it.

But instead of, you know, invading yet another country on the barest of pretenses, how about legalizing the drug trade, thereby depriving these cartels of their funding and, indeed, their very raison d'etre. The addiction problem you can solve within the health care and education systems, at far less cost (and much more effectively!) than waging a global, never-ending war on drugs.

Then, if you're up for it, after the members of the now-defunct cartels are weakened, you can go after them with the bullets. Should be much easier.

I mean, you say it right in your post: "long, ugly, and bloody for all parties involved". In fact what you describe is not a hypothetical at all, it is the existing War on Drugs. But then you just sort of dismiss that and continue on with your insane 'plan'.