I think the C's rebounding woes come from a few different areas.
First, they really don't have many great rebounders in the frontcourt. While KG, JO, and Shaq were always good to great rebounders when they were younger, they have all lost at least a step or two, so they aren't what they used to be. And Davis, Green, and Krstic are all below average rebounders for big men.
Second, the system is not built for big men rebounding. They don't do a ton of offensive rebounding, mainly because they emphasize getting back on defense so much, and defensively, the system pulls the big men away from the basket so much in order to help on the perimeter, which leaves them out of position to box out the big men who might be rolling and crashing the boards.
Third, the system relies a lot on the perimeter guys crashing down and grabbing a lot of rebounds (which is why the C's were at their best last year when Pierce and Rondo had a lot of rebounds). This is compounded by the fact that their PG happens to be at his best when they can get out on the break, which means the perimeter guys need to leak out early to get behind the defense, leaving the C's vulnerable on long rebounds.
Now, in game 2, I think we saw a perfect storm of all of these factors combining to really kill them. The personnel issues are obvious, but they were even worse than normal because of JO being in foul trouble, as well as the Knicks matchup forcing them to go smaller for long portions of time with Green at the PF spot.
They also were taking their help defense to the next level later in the game by using Davis as essentially a roamer to double-team Melo anytime he got the ball, which left him completely out of position to block out his man once the shot went up, and he is not big enough to overcome that (I also think this defensive strategy was part of the reason we didn't see JO late in the game, because Davis has quicker feet to blitz and get back to defend).
And finally, the C's perimeter guys were leaking out more than normal in the hopes of righting the ship offensively. There were many times when Rondo, Pierce and Allen were way behind the defense, which means they were not going down to help on the boards...which was especially bad since NYK were launching a lot of 3's, leading to long rebounds, which are always the perimeter players domain.
Anyways, I don't think anything is going to change about the personnel. We had similar issues last year even with Perk, and this team just are what they are, but there are some changes Doc can make to improve their rebounding. The tough part is going to be how to do that without sacrificing the defense, or the offense, which was finally showing some life last game.