The NYT crossword has a steep difficulty curve increasing over each week. It took me a few years of regular solving before I could consistently finish a Saturday puzzle. If people are only seeing the NYT puzzles on weekends, no wonder they think it's elitist. Try a Monday puzzle!
(Also, the obscure 1930's opera references have been on the decline since Will Shortz took over for the late Eugene T. Maleska. Maleska was an admitted elitist who filled the puzzles with high culture, while Shortz prefers misleading wordplay. You see a 5-letter clue, "Pentagonal part of a diamond", and it takes a while for the baseball reference to click - but once it does, PLATE is the obvious answer.)
I've looked at the USA Today crossword a few times, but found the quality relatively poor. Lots of dull clues and obscure three-letter words to make the fill fit together. No wonder he got away with plagiarism for so long, the people who'd notice wouldn't bother sticking around long enough to find anything wrong.