It's even harder to understand given that the the language of the state consititutional provision was changed, by vote of the people, in 1986. What was added was "for security or defense of self, family, home, and others, and for lawful common defense, hunting, recreational use, and all other lawful purposes,".
I don't live in Nebraska (not since 1977), so I don't understand the problem either. I did a little checking and found only one state Supreme Court case addressing the issue since the changes were made. The Court basically ignored the changes. However it was "bad case" as the defendent was not a "nice person", but rather a penny ante criminal. AFAIK, the laws, including the requirement to have a FOID to purchase a handgun, which was passed about the same time (can't recall if before or after) the constitutional amendment, have otherwise not been tested at the State Supreme Court level.
With that constitutional provision, Nebraskan's shouldn't *need* a CHL to carry a gun, concealed or otherwise.
As I said, I don't live there, but my parents, brother and in-laws do. I did get stopped for speeding once in the mid to late 1980s on the way from my in-laws farm to their house in town. I had enough guns in the trunk to start a revolution in a small central Amercian country, because we'd had a family picnic and shoot that afternoon and I had everybody's guns) But the subject never came up with the county mountie. (Who my father in law said is a known Cylmer :) )