I felt Legend of Korra, especially Book 1, questioned the importance of the Avatar and his/her role in society quite a bit.
Right off the bat, the existence of Republic City stands in defiance of one of the core obligations of the Avatar, the preservation of the four separate nations/cultures. Republic City is established when Zuko, and eventually Aang, decide that the benefits of cultural exchange and integration could outweigh the benefits of separation, decades before the events in Legend of Korra even take place. It's an enduring testament to the idea that the values the Avatar embodies are at worst misguided, and at best worth calling into question. That's Legend of Korra's primary setting.
On top of that, many of the societal conditions that would have necessitated the intervention of an Avatar have either become irrelevant or are now being addressed by villains *and* allies who now have the ability and determination to intervene in matters that would have previously been the responsibility of the Avatar alone. For example, there's no need for the Avatar as a spiritual leader when spirits no longer have a significant presence in society. Who needs the Avatar to be a global keeper of peace and balance when the Air Nomads can handle the job for the most part while Korra is incapacitated? In the cases of the anatagonists, Amon and Unalaq take it upon themselves to encroach upon the Avatar's duties after recognizing the failings of previous Avatars' visions for the world. While those two characters eventually turn to outright villainy, the show (and Korra herself) eventually treats their respective causes as correct and even noble. Hell, Unalaq's desire to undo the separation between the human world and the spirit world literally revives a culture that had been all but erased from the world due to genocide. And his ideology directly contradicted the wisdom of the very first Avatar! If that doesn't blatantly support the idea that maybe the role of the Avatar isn't as vital as it once was, then I don't know what does.
Also, Amon's Lieutenant explicitly states that the Avatar isn't necessary anymore.
And all that's just off the top of my head. I could probably come up with even more stuff that would support this as a recurring theme if I did another re-watch.