First and foremost, anyone thinking critically while reading Mockingjay or after reading it would simply need to recall the formation of the rebellion that takes place in the book(s) rather than the events that happen toward the end of it. President Coin declared herself leader of the REBELLION despite the obvious fact that it was KATNISS who ignited, fueled, and led it.
Second, Coin never gave Katniss control over her image as the Mockingjay--even when Katniss acted on her own accord, Coin ALWAYS fabricated Katniss to suit HER intentions for the rebellion. It doesn't take much to realize that all Coin wanted was authority--I wasn't even surprised when Katniss killed her at the end because of the various allusions to her tyrannical dark side that were so subtle but "out there" in her personality, especially when her and Katniss interacted (it was never pleasant).
In essence, Katniss and Coin did not see eye-to-eye regarding the rebellion. Katniss' lack of conformity was the distinguishing factor. This is why Coin wanted Peeta rescued at the end of Catching Fire because he conformed to the Capitol and made his and Katniss's affair seem convincing--Katniss couldn't achieve that, and President Snow threatened her for it. Even Haymitch admitted that Peeta was more simple to be around than Katniss because he could be persuaded and make things go smoothly (Remember the events at the end of The Hunger Games and during the first half of Catching Fire?). There is even a moment in the novel during a broadcast in which Peeta, still his normal self, attempts to get through to Katniss by telling her that she is a pawn being used to fulfill another regime. Boggs even told Katniss that Coin (without specifically saying it) was afraid that Katniss would rise to power on her own without even trying. To counter Katniss, Coin performed numerous tasks to suppress her including sending Peeta with Squad 451 in the hopes that he would kill her in a fit of unexpected rage.
The bombing of the Capitol children and Prim wasn't to break the last of the allegiance the people of the Capitol felt to Snow, but to mentally unhinge Katniss so that she would place her entire faith in Coin. And Coin nearly succeeded given that Katniss's depression and desire for vengeance briefly consumed her. Snow's last conversation didn't even need to be forced so that he could get through to Katniss; he knew how intelligent she was. All she had to do was put the pieces together to understand just how much worse Coin was in contrast to Snow. But even then she was trying to find a way to deny the truth because she was overwhelmed with sadness and anger...so when Coin suggested a "final" Hunger Games using the children of the Capitol's elite families, it solidified the truth in its entirety.
Coin made everyone fight a battle only to repeat an aspect of an atrocity they originally rose up against. A sane person like Katniss, Beetee, Haymitch, Peeta, and Annie knew it wasn't right despite the fact that the people of the CAPITOL enthusiastically watched the Hunger Games each year with no objection. Coin's plan wouldn't pacify those who directly suffered despite her claims that it would--there would never be any true compensation for the those who were lost in the games, the first war, and the second war. And, had a Hunger Games been hosted with Capitol children, Coin would have likely used another excuse, that the people were seeking more vengeance...it would eventually paved the way for her to embody and take over Snow's position. There would have been no change.
Katniss voted in favor of the games for one reason, and that was to gain Coin's trust. Her bargaining for the role as Snow's executor was simply a facade to cover up her own intentions. Katniss only had one chance to kill Coin and if she ruined that in any way...nothing would have changed.