I will say that Advent Children seemed to bank on some things from Crisis Core as well. While Final Fantasy VII was intended to stand alone, really you have to view it almost as a trilogy to get the whole story and exposition. Also I think some of the unanswered questions that still remain are going to be covered in the prequel to Crisis Core about the Turks.
To give a simplistic answer of a couple of things I noticed when reading through the review:
#1: At the end of Final Fantasy VII it left things fairly open as to whether humanity even survived, since they were to be "judged" by "Holy" and on top of this "Holy" intervened too late to undo ALL the damage.
Apparently humanity was judged worthy, but the events left a scar in the world and the spirit of Jenova and her champion Sepiroth wound up in the lifestream of the planet. This infection in the world was causing the Geo Stigma, and the three bad guys were basically avatars of the taint who were acting to bring Sepiroth back so he could finish destroying the planet, towards that end they need a specific macguffin to gain enough power.
#2: Cloud is functioning on Angst because of the death of Aeris and the fact that he pretty much saved the world, but wound up making it pretty miserable even with his victory. What's more he's infected by Geo Stigma himself and is dying from a slow waste.
Part of why this concludes the story is the bit with his death and resurrection (or near death) at the end which allows him to meet Zack and Aeris (who were in love) and make peace with what happened so he could finally be happy with Tifa.
Really him being miserable makes sense, and honestly it was intended to kind of mirror the events of Final Fantasy VII. Cloud is simply put the baddest guy on the planet pretty much, even if he was never a SOLDIER, his problems have always been mental. He didn't quite become the "wimp" he was in Crisis Core which would have made no sense, but the point is that he was losing (or breaking even) in the first fights because of what was going on in his mind. Eventually he gets over it and does what he needs to, and pretty much trashes everyone in a rather impressively animated series of fights, one battle after another. It wasn't until after he was shot in the back that he really comes to lasting peace however.
When I say Cloud is the baddest guy on the planet, it should be noted that none of the other heroes like Zack could get past Sepiroth. While the "Last Order" anime and "Crisis Core" kind of ruined the scene from Final Fantasy VII (IMO, I didn't like the retcon, though I understand why they did it that way to put a greater focus on Zack) it's noteworthy that he manages to pretty much take him down even after being impaled. Later when he has Jenovah enhancement it should be noted that Sepiroth doesn't generally take him on head to head until the end, and then he loses in a straight one on one fight. The overall score basically being Cloud 3, Sepiroth 0.
#3: The Turks were never really villains I don't think, they merely seem that way because of the character's perspective at the beginning of Final Fantasy VII. In reality they are quite honorable, and doing what they think is right. What's more they treat Aeris well, and have actually protected her, Hojo being killed by Sepiroth for that reason.
The whole thing is that "Shinra" isn't a straightforward bad guy, even in an evil corperation sense even if Barret sees it that way.
As far as being comedy relief, there is some truth to that in the exact sequences, but conceptually what they are doing is actually fairly heroic. They are basically fighting against opponents totally out of their league with Jenovah enhancement, despite having no chance whatsoever. For all of the humor involved, I think they conveyed pretty well that The Turks are far, far tougher than any ordinary people around, but they are simply dealing with enemies dramatically outside of their power range at this point.
Generally speaking Tifa, Vincent, Yuffie, and Cloud are the only characters that are strong enough to be a serious factor. Of them Yuffie was the only one who was overlooked (quite probably because the whole Pagoda of the Gods sequence was optional). Tifa was trained by what was basically the world's greatest martial artist who left her his ultimate techniques, I don't even what to think about what she could do with Jeonovah enhancement. Indeed the biggest joke in the movie was when she whooped up on one of the bad guys and then the victory music from the game plays, the guy gets back up (it was his cell phone ring tone) and takes her out in a cheezy fashion to advance the story. A brilliant homage/satire on this trend in JRPGs (you won the fight! but now you lose anyway in the cinematic!). Nobody has gotten into the exact nature of Vincent's enhancements, but he was formerly the leader of The Turks (which is why he apparently rescued Elena). Perhaps if it's released the "Turk" game I've heard about will answer that... or maybe it's in "Dirge Of Cerberus" which I have not played.
Well, enough Final Fantasy VII rambling, my point is that I think some bits made a bit more sense than you might at first think.
That said Advent Children wasn't a great movie, it does not stand on it's own merits even a little, and as beautiful as a lot of the visuals and fights are, I can't say it tells a really great story either. If you think of it purely as an animated epilogue to Final Fantasy VII I guess it's a bit better.... however you still have to be a real FF VII nerd.