Babcock is big on out of game effort. He's said before that if you aren't playing, and want to play, work your ass off in practice, be in the weight room, and constantly put in effort.
Leivo doesn't appear to have that trait to him. Which is fine if you are an established player, but Leivo is a guy trying to get playing time, and obviously that isn't there if Babcock isn't playing him.
I think it was more of a stylistic issue. He was not a PKer. That was the main difference between him and the guys who beat him out for playing time. The Leafs were not short of offensively capable guys as it was, and while Leivo is solid defensively, he is not going to beat out Brown or Kapanen in that area either. You also have to take into account that it was not like Leivo was getting passed up by one guy after another. Really, it was only Johnsson and Kapanen, who were two of the Leafs' top prospects + played the defensive game Babcock likes out of his younger guys.
I can not comment on his effort in practice, but I would imagine he is not slumping in that area based on his play on the ice. Always battled hard and was very engaged. It just seems like there were not enough spots for pure scorers, especially when the Leafs did not suffer many injuries. Between the top 9 outside of Matthews (Hyman, Kadri, JvR, Marner, Nylander, Bozak, Brown, Marleau), there were only a total of 4 man games missed. Only one of those games was from a winger (JvR). Babcock is not going to sit any of those top 9 guys in favour of Leivo, and Kapanen and Johnsson were the only two people who really deserved to play over Leivo so they did. It's not like someone else came in and took ice time away from him. Between the two guys who were rightfully ahead of him and the lack of injuries, Leivo (and Fehr and Soshnikov) did not have many opportunities. Somehow, he did still get into 16 games when there were only 24 games missed by guys he was realistically capable of replacing. However, with Martin and Komarov already replaced by Johnsson and Kapanen and Bozak obviously being replaced by Tavares, there was nobody new brought in really to replace JvR (outside of Ennis). That's where Leivo's opportunity comes in. He can play the JvR spot on the PP and he can provide a big body with offensive skill that was lost when JvR left. Ennis is an underwhelming small forward who will need to do a lot to prove he is worth any sort of NHL shot, never mind a consistent one.
When it comes to Leivo, I am not worried about the past. I do not think there is any sort of grudge on the part of Babcock, just no opportunities for him to play. Now, he has been given his opportunity and the spot he has wanted for two years. It's not up to him not to waste it.