You've misunderstood a few minor aspects and one whopping big major aspect of the ending and moral of the Hunger Games series. I'll get the most important point out of the way first.
The 76th Hunger Games with Capitol children almost certainly never happened.
To understand this, we have to consider what Katniss's motivation was for supporting Coin's plan. Let's have a look at the relevant passage again:
Was it like this then? Seventy-five years or so ago? Did a group of people sit around and cast their votes on initiating the Hunger Games? Was there dissent? Did someone make a case for mercy that was beaten down by the calls for the deaths of the districts’ children? The scent of Snow’s rose curls up into my nose, down into my throat, squeezing it tight with despair. All those people I loved, dead, and we are discussing the next Hunger Games in an attempt to avoid wasting life. Nothing has changed. Nothing will ever change now.
I weigh my options carefully, think everything through. Keeping my eyes on the rose, I say, “I vote yes…for Prim.”
“Haymitch, it’s up to you,” says Coin.
A furious Peeta hammers Haymitch with the atrocity he could become party to, but I can feel Haymitch watching me. This is the moment, then. When we find out exactly just how alike we are, and how much he truly understands me.
“I’m with the Mockingjay,” he says.
-- Mockingjay, Chapter 26 (emphasis mine)
First of all, note that this is pretty much the only place in the entire trilogy where Katniss's thought processes aren't described in detail. Usually we know exactly what she's thinking and why she's thinking it - if anything, these things will be overexplained. But here, the author suddenly hides Katniss's inner monologue from us, telling us only that she "weigh[s her] options carefully" and "think[s] everything through", which is about as vague as you can get.
That's already enough to hint to us, the reader, that her motivation is something more than she's showing. Then there's the "Keeping her eyes on the rose" bit, which suggests she has something to hide - she doesn't want to meet Coin's eyes. And finally, her interaction with Haymitch. This is the moment when "we find out [...] how much he truly understands me". If she doesn't have anything to hide from the others in the room (specifically Coin), then why not state her motives aloud rather than relying on Haymitch's deep understanding of her?
This, then, is the moment when she decides to kill Coin.
It fits with the rest of the chapter. After her encounter with Snow in the conservatory, Katniss spent several pages agonising over whether it was the Capitol or District 13 who caused the death of her sister. She tried to go to Haymitch for advice, but ended up walking out instead. And right here in this meeting, she comes to the final, despairing, realisation that Coin is hardly any better than Snow. Which of them it was who dropped the bombs on the children hardly matters; it's now 100% clear that Coin is the kind of person who would do such a thing.
So why does she agree to the final Hunger Games with Capitol children? Well, given that she's already decided to kill Coin, it doesn't really matter which way the decision goes; once the new president is dead, everything is up in the air again, especially since she's going to die before publicly announcing the new Hunger Games. Katniss's only goal now is to keep Coin unsuspecting, and the best way to do that is to agree with her suggestion.
Katniss doesn't support the idea; she only agreed in order to lull Coin into complacency.
That deals with your first bullet point. The rest are easier to answer.
Katniss does indeed shoot Coin because she's come to the conclusion that Coin is just another Snow. Panem under Coin might have been little better than Panem under Snow. But that doesn't mean the actual Panem seen at the end of the series is so bad. Both Snow and Coin are dead, their reigns over.
Snow escapes execution, yes, but it's unclear whether he died by choking on his own laughter or by being crushed by the angry crowd. Either way, it doesn't make much of a difference. He died a humiliating death, not a glorious one.
Many characters died who might not have done if it weren't for the revolution, yes ... but if it weren't for the revolution, their lives would have had little meaning anyway. They would have been, as Peeta so famously put it, just pieces in the Games (literally or figuratively: even people not involved in the Hunger Games were nothing but pawns to be manipulated by people like Snow and Coin). The brutality and inhumanity of war is certainly a theme in the HG series; its futility, not so much. Which leads me on to the final point ...
Life did get better in Panem at the end of the series.
I've already written an extensive analysis of this elsewhere. The incorrigibly upbeat yet endlessly cynical Plutarch Heavensbee gives a mixed assessment of the future:
"Are you preparing for another war, Plutarch?" I ask.
"Oh, not now. Now we're in that sweet period where everyone agrees that our recent horrors should never be repeated," he says. "But collective thinking is usually short-lived. We're fickle, stupid beings with poor memories and a great gift for self-destruction. Although who knows? Maybe this will be it, Katniss."
"What?" I ask.
"The time it sticks. Maybe we are witnessing the evolution of the human race. Think about that."
-- Mockingjay, chapter 27
Now Plutarch is not necessarily to be believed or trusted, of course, so from an in-universe point of view, this doesn't tell us or Katniss a great deal. But if we interpret this as the author speaking to us, the readers, through her characters, it becomes a sort of brief analysis of the outlook on the future of Panem. Perhaps things really will get better this time, or perhaps it's just a brief lull before another downhill slide of the society. Only time will tell.
And once time has passed, or at least a few years of it, the improvements in people's lives are apparent and enormous. They work together, using both Capitol technology and their own physical abilities, to make Panem a decent place to live. Instead of slaving their lives away in the mines, the population of New Twelve are preparing to build a medicine factory:
We're not alone. A few hundred others return because, whatever has happened, this is our home. With the mines closed, they plow the ashes into the earth and plant food. Machines from the Capitol break ground for a new factory where we will make medicines. Although no one seeds it, the Meadow turns green again.
-- Mockingjay, chapter 27
And going even more years into the future, by the time Katniss and Peeta's children are growing up, not only are the Games merely a distant memory, but the old way of life is so far removed from the new era that Katniss fears her children will be "frightened to death" even to hear about it:
The questions are just beginning. The arenas have been completely destroyed, the memorials built, there are no more Hunger Games. But they teach about them at school, and the girl knows we played a role in them. The boy will know in a few years. How can I tell them about that world without frightening them to death?
-- Mockingjay, epilogue
So it seems clear that the new Panem really is a more pleasant and prosperous place than the old Districts, while presumably being less comfortable and cloistered than the old Capitol.
Coin would have been another dictator like Snow, but she was killed and Paylor apparently turned out to be a more benevolent leader for the new Panem.
Further reading:
