This is a poorly researched article. The Mexicans hunt deer and I know several with pump and bolt action rifles of other than military calibers. That would mean no 7mm Mauser, .30-06, 7.62 Nato, or 5.56 mm. These are not hidden in the closet. The point about Americans travelling to Mexico is well advised.
I suspect the author has never travelled outside the tourist areas of Mexico. I've seen a civilian on the back of a horse openly carrying a hunting rifle not far from the southern border of Mexico.
Guns may not be carried in public, either openly or concealed.
They shoot pistols too. The .38 Super cartridge is very popular in Colt 1911-style automatic pistols down there, because it has never been a military cartridge and therefore (unlike the .45 ACP) is legal -- with considerable restrictions to be sure, but pistols can be possessed with the right permit.
-ccm