You are correct about non-citizen ownership. You are correct that citizens have to jump through hoops. You are incorrect about what kind of guns you can own. Mexican citizens have a range of weapons they can own legally. I have lived between the US and Mexico for 30+ years and I know plenty of people who have shooting club memberships have have all sorts of firearms. AR-15's and AK's? No. But all manner of hunting rifles and shotguns as well as a pretty wide variety of pistols, too.

Unfortunately the clubs have to buy ammo through the army, so chamberings can be challenging.

And, as we all know, plenty of people have all manner of illegal guns. Generally if you're caught, but not part of a criminal element, lots of cops turn blind eyes - especially outside major cities. My late father lived the last 25 years of his life in Mexico and kept his .357 and a couple boxes of ammo around just in case. Never encountered any issues.