I'm not sure what a "narco-state" would be in this instance. That Mexico is massively corrupt, there's no question. That it is incredibly violent, that is certain. That the crime gangs are involved in drug trafficking is obvious.From my reading, the narco gangs have scored a few high profile successes and are adept at picking off isolated cops and government functionaries but usually get the silver when federal forces arrive in strength.
I'd say it's more like a large version of South Armagh in the Troubles than an existential threat to the Mexican state.
I wouldn't however conclude that all their problems are down strictly to drug traffickers. Mexico has a long history of violence and instability. They were ruled by a sham democracy in a one party state, although that is beginning to erode into something more closely resembling an actual democracy. I say "beginning to", because they have a long way to go. There were few complaints from the usual suspects about the state of democracy in Mexico as the Mexicans knew enough to stay within their prescribed limits in the American "Near Abroad".
If you're comparing it to South Armagh though, then I'm not sure that you really understand the flavour of it. Mexico had a murder rate of 25 per 100,000 in 2017. That makes them as dangerous as say, Washington DC. That's quite frightening, isn't it? And of course the drug gangs in Mexico sell their drugs on to the well armed drug gangs in the US, who then retail it to their buyers at a considerable mark-up.
I've been to Mexico on business. I wouldn't choose to go there for a vacation though. Crime is a problem, and the once relatively safe tourist towns are no longer so. Things might be better if you go on a package vacation that takes you straight from the airport to the resort and stay there until you return home.
My experience on business though has been that Mexico City is vast, surprisingly so if you aren't familiar with what to expect. The climate is quite mild due to the high altitude. However, the air was massively polluted when I was there. The sky was yellow and you could only see a limited distance in the persistent yellowish fog-like pollution. I'm sure there's some fantastic history and culture to be seen there, if only you could actually see anything.
The industrial cities are a bit less inspiring. The ones near the American border are appalling, the ones further from the border less so.
I spent some time in one paritcular mid sized industrial city. There were lots of nice new looking divided highways without much traffic on them, and spanking new factories and hotels oriented towards businessmen. There was money there, but not much seemed to trickle down to the average Mexican. Pretty much everything that was worth anything seemed to be owned by the Americans. This is perhaps the reason that the American government don't seem to get as worked up over democracy and human rights in Mexico as they do with elsewhere in the world.
The Mexicans that I met were quite friendly, cheerful, and pleasant. But then again these were factory workers and engineers, not gangsters. One engineer knew that we were Canadians and sidled over to me at one point to whisper quietly, asking if I was aware that the Americans had stolen half of their country. He was genuinely shocked when I told him that the Americans had tried the same with us and gotten a good shoeing for their pains, that being one of the advantages of being part of the British Empire at the time as opposed to a small independent republic. They haven't forgotten their history with the US.
Mexico is a country of 130 million people. That makes them half again larger than for example Turkey. Total nominal GDP is about $1.3 trillion, which puts them as the 15th largest economy in the world. If you measure it using PPP standards, which takes into account the cost of living, then they are number 11, ahead of Italy and within shouting distance of France (IMF estimates for 2019). Even if we just stick to nominal GDP though, they are still quite significant.
Economic growth is erratic. They will show a few years of good growth, followed by several years of stagnation. The potential is there, but the result is nothing compared to what has been seen in east Asia.
Mexico have a new president. He is a genuine change compared to what has come before. Whether or not that is a good thing remains to be seen, but he is promising to really shake things up. As I said before not much of what wealth there is seems to trickle down to the lower ranks of society, but he intends to change that.
This development is not necessarily welcomed in Washington. I won't be surprised to see the American political and political pundit class to develop a sudden deep concern for freedom and democracy in Mexico in reaction to this.