One criticism I have of him is he doesn't seem concerned about rebounding or defensive big guys. AJ was supposed to fill that role, but he hasn't been consistent. It was not a good sign.
It all started when they didn't pay Perkins, who was their defensive rebounding big in '08. A guy who took a lot of heat off KG. They weren't the same after that, though it was mixed in with KG's injury. The other guys they let go for $$ were Powe (offensive rebounder supreme) and Posey, defensive 3P guy. When KG got hurt, they were that much weaker because those other guys weren't around.
OK, other teams grossly overbid for these guys, but they didn't seem to make a strong effort to keep them. I always thought Wyc was the controlling factor--he figured he got his championship, and he wasn't going over the cap to get another. Plus maybe there was an under appreciation of what a defensive guy like Perk brought to the team. I don't think Wyc was in touch with Celtics history, and the importance of Bill Russell in winning championships with his defense. He just didn't want to pay a defensive guy. Maybe still doesn't, though AJ did not come cheap.
Perk, God love him, was a straight scrub after his knee injury. Ainge dumped him for some value before that became totally apparent to anybody who watched him play.
Not just that, but the league has mostly passed these guys by. Being a great defensive big man has a lot more to do with the ability to switch onto smaller players without sacrificing too much on the boards, as opposed to the ability to defend 1-on-1 in the post with a really high block percentage.
It's true, Ainge doesn't seem to place a great deal of value on the "Volleyball Player" archetype -- e.g. big pogo sticks who have very little skill but are great at swatting away or grabbing and dunking the basketball within 5 feet of the basket. The only time I know of that Ainge drafted a player like that was Fab Melo (RIP, big guy), and I think the hope with Fab was that he was a bit of a tabula rasa and could develop into more than just a presence near the basket.
I think the aversion to that sort of player is defensible, though. Players of that sort tend to be very expensive for what they are, and I think there's an argument to be made that unless you have some truly superlative offensive creators elsewhere on the floor, the limitations of that kind of player (typically they can't really pass, shoot, defend the perimeter, or hit free throws) are almost as much of a drag as their strengths are a boon.