For those who take issue with the NAB, what bothers them the most is the introductory notes and footnotes. Sure, they may be irked that Luke 1:28 says “Hail, favored one” rather than “Hail, full of grace”. But it is the extra stuff they find most problematic (in my experience, at least).
Mainly, this is because the notes put forth some of the fruit of the historical-critical method of biblical scholarship as though it were established fact. And some of those conclusions fly in the face of more traditional interpretations.
For example, when it comes to authorship, the more traditionally minded tend to want to say that the books of the Bible were written by the person whose name has been connected to it for thousands of years. So if it is a letter that says it was written by Paul, that means it was a letter from St. Paul. In the NAB, they will bring up those issues about authorship and primarily present just the one side that casts doubt on the traditional understanding.
Other footnotes and such spend so much time analyzing the “origin” of this or that phrase that it might lead someone to conclude that things didn’t really happen as indicated in Scripture. This is a point that can be made when properly understood, but I think the concern is that many people who would read them would walk away with the notion that we cannot really trust much of what is written in the Bible as being “authentic.”
Honestly, I’m not a huge fan of the footnotes either, but I do think the criticism gets overblown at times. Any translation will have strengths and weaknesses. If you come across something that doesn’t sit well, then you investigate other sources.