Idina Menzel had the privilege of singing one of the biggest songs of the year on the 2nd biggest album of the year. However, it may have turned out to be a curse. Menzel clearly struggled with her performance on New Year's Eve, and at her performance at the Academy Awards in February. This is not a one time thing, it is evident of a real problem.
Now we all know that it was cold in New York that night (snow can be seen falling all around her), but while this would be an excuse in most cases - at least Beyoncé would think so - Idina also butchered her performance in an air conditioned theater in February. Both performances were screechy and pitchy, and it's hard to give her a pass for either performance. Yes one performance was stressful on the vocal chords themselves and the other on nerves, but audiences don't care. In this business, you make it work, or you fail. It's a tough rule, but a Broadway icon like Idina should know and understand that by now, and simply make things work.
The real problem here is a strange one, Idina is not mixing her head and chest voices at all. She's simply screaming and screeching in every performance, hoping that pushing a lung's worth of air forward will sustain her higher notes. This is not good technique, and very strange for a Broadway performer since mixing is almost a requirement for musical singers.
All of that being said, Idina has had decent performances of the song IE her Jimmy Fallon performance, though lowered a key and still showing some fatal vocal practices. Idina has some things to work on and consider for the future, including hitting more than 75% of her notes.