I agree with the sentiment that Danny didn't do enough to help the team in the offseason following the championship. The rest of the league got better while we actually got a little worse.
Regarding specific moves, I generally don't like making judgments of a GM's moves in hindsight, but drafting JR Giddens in the 2008 draft has to be considered a mistake. If you look back at that draft, it was extremely deep with many very good players. Giddens was one of only five or six "busts" in that first round. There were also several very good players who dropped into the second round, including DeAndre, Asik, Pekovic, Chalmers, Mbah a Moute and Dragic.
Even then, you can see what Ainge liked. Giddens was a guard with good size and elite athleticism. He flashed an ability to do a lot of things well, but unfortunately could never do them consistently.
He could have been a SG/SF version of Bradley, but unfortunately he never adapted to the NBA well enough. Looked like he never shook the nerves of playing on the biggest stage. Possibly wasn't smart enough to pick up the system either, though that's admittedly a guess on my part.