In the final event of the day, Kennett bombed out of the elimination race, ending his chances at an Olympic medal.
Elliott called out Kennett on his efforts.
"Dylan's got to learn how to be a champion and it's not by giving up," Elliott told Radio Sport.
"I think we've just seen someone who hasn't been able to focus for the elimination and do what he is really capable [of]."
Craig Kennett thought the comments from Elliott were appalling.
"What great support from your leader. It's just a disgrace to call Dylan a quitter," he told Fairfax media.
"Mr Elliott said that Dylan needs to learn to ride like a champion but he was a world champion last year.
"What has Mr Elliott been a champion at to know how they handle setbacks? Tiddly winks?"
Craig Kennett said his son was critical of the Bordeaux track the team prepared for the Games on, because it wasn't like the tracks they raced on.