Organization XIII was a really interesting concept for a villain group, but I don't think their role in Kingdom Hearts II worked out very well. Even though Kingdom Hearts II is a fairly long and involved game with numerous stages and boss battles, the main villains of the game hardly got any screen-time or character development. As a result, it was hard to remain interested in them as characters. In order to have been good villains, the various members of Organization XIII should have appeared before Sora much more regularly than they did in the game.
The weakness of Organization XIII in Kingdom Hearts II is particularly glaring because Organization XIII put in a very strong appearance in the GBA game Kingdom Hearts: Chain of Memories. First off, the four members of Organization XIII from Sora's story in Chain of Memories appeared regularly in the games plot from the get-go. At the end of every floor of Castle Oblivion, the members of Organization XIII either appear before Sora to talk to him or talk about their plans with each other in a cutscene away from the main characters. They also regularly appear as opponents: each of the four Organization XIII members fight the player twice over the course of the game. As a result of all of this, those four members of Organization XIII leave a strong impression.
Sadly, the seven (or eight if you count Roxas) members of Organization XIII who feature prominently in Kingdom Hearts II don't have nearly as much stage presence as the original four (which is rather ironic, since Axel appears in both games). A great deal of this comes from how seldom many of the Organization XIII members appear individually. Other than Axel's two-part appearance in the Roxas prologue, the player first encounters Organization XIII when all six of the remaining members appear as a group to taunt Sora. After that, only Axel and Saïx reappear on a regular basis. Here is a basic summary of the appearances of all of the members of Organization XIII:
1) Puts in a short appearance in Olympus Coliseum (1st) (not even as a big part of the world's plot) where he mostly runs away and acts cowardly. At least he forces the player to fight a short minigame-type battle. Unmasked.
2) Appears out of nowhere during mid-game Hollow Bastion event. Boss battle and dies, having never played a role in the game's overarching plot or done anything significant in any of the Disney sub-plots.
1) Plays mindgames with the Beast in Beast's Castle (1st).
2) Helps carry on the plot of Beauty and the Beast(2nd). Unmasked, boss battle and dies (in really short succession), having never played a role in the game's overarching plot.
1) Taunts Sora with mysterious words during the original Organization XIII appearance in Hollow Bastion. However, since he keeps his hood down and doesn't reveal his name, the whole time he just looks like a generic member of the Organization, so it hardly counts.
2) Shows up in the Land of Dragons (2nd), where he has one line of dialogue and no major story role. Unmasked.
3) Appears out of nowhere in the final dungeon. Boss-battle, dies, having never played a role in the game's overarching plot or done anything significant in any of the Disney sub-plots.
1) Shows up to brilliantly outwit the party in Port Royal (2nd) and set up some challanges. Unmasked.
2) Appears through a Megaman-style fortress boss door in final dungeon. Boss-battle, dies, having never played a role in the game's overarching plot.
Axel (this is where things improve somewhat):
1) Appears before Roxas in the prologue during the Struggle match. Mini-boss battle.
2) Shows up to have a major battle with Roxas at the end of the prologue.
3) Appears in several scenes alongside Saïx in and around the mid-game Hollow Bastion event as a means of bringing Kairi into the plot.
4) Comes out of nowhere to fight alongside the heroes against the horde of Nobodies en route to final dungeon. Sacrifices himself heroically. Dies.
1) Shows up alongside Axel in various scenes around the mid-game Hollow Bastion event as a means of bringing Kairi into the plot. He performs most of the major exposition of Organization XIII's plans here. Unmasked.
2) Reappears in numerous cut-scenes in the final dungeon, helping move along the Riku and Kairi plot and doing basic exposition.
3) Appears through a Megaman-style fortress boss door in final dungeon. Boss-battle, dies.
Xemnas (the leader):
1) Appears briefly in the mid-game Hollow Bastion event, and in a few associated flash-backs. Unmasked.
2) Reappears in final dungeon after all of the other Organization XIII members have been killed. Talks melodramatically with the party a little. Chain of final boss battles, dies.
In summation, only Axel and Saïx have any major recurring appearances across the game, and even those are limited to a few discrete sections of the game (near the mid-game Hollow Bastion event and the final dungeon). Saïx is also the only member of Organization XIII to regularly interact with other members of the Organization. He is also used to do all of the plot exposition. The rest of the members often feel like they are really only filling out a necessary quota.
The biggest problem with how the Organization XIII members are used is that they never interact with each other in any meaningful way. In Chain of Memories, the four members of Organization XIII interacted with each other constantly, engaging in a complex web of alliances and betrayal. Those interactions established their identities and built an exciting basis for the game's plot. The Organization XIII in Kingdom Hearts II would have been much more interesting if it was full of sub-factions and individual members pursuing their own agendas. Without complex interactions and individual motivations, most of the members of Organization XIII end up acting like a random collection of generic villains with themed super-powers.
The other big problem of Organization XIII, their lack of on-screen appearances, could have been remedied in a number of ways. One of the biggest would have been to make the Organization as a whole more prominent in the Disney Worlds. While Xaldin has a big role in Beast's Castle, and a few of the others make minor appearances in others, the developers could have gone a lot further in integrating the Nobodies into the Disney stories. For example, Demyx or Luxord could have easily replaced Ursula in Atlantica's plot, which would have made a lot of sense, considering that Ursula was defeated in the last game and returns inexplicably in Kingdom Hearts II. They could have also put in a more prominent appearance in Agrabah, where the game does mention that a Nobody interfered but never shows it happening.
In addition, having the player fight various members of Organization XIII multiple times would have improved things a lot. For example, introductory boss battles against Saïx or another Nobody in Twilight Town or at the end of the big mid-game event would have been appropriate. Having a second boss battle against a villain elevates that opponent from one-shot boss to recurring character in a players mind. For example, Black Waltz no. 3 from Final Fantasy 9 is a very memorable opponent because of its persistence, even though it doesn't have any kind of major story role. In any case, more Organization XIII fights wouldn't have hurt, since they were generally the most exciting battles in the whole game. Maybe have a few of the members transform into those giant Nobodies like Marluxia did at the end of Chain of Memories?
If you create a large group of villains instead of one or two, you need to do something different in order to make them work. You have to do more than just rely on the threat of the whole group, but instead sell all of the individual members as credible characters in their own right. Giving them all consistent appearances and strong individual motivations goes a long way in achieving that. If the developers of Kingdom Hearts II had stuck closer to what worked so well in Chain of Memories, Kingdom Hearts II would have had a much stronger cast of antagonists.