Obviously the aim of her mission was to Kill Snow - no matter how desperate or unlikely - however, I suspect the only real outcome of her adventure was that she witnessed first-hand what happened to Prim, and set Katniss off on the idea that President Coin can't be trusted.
Let's break this down
This is a pivotal plot point of the series; Katniss is being used by
both sides of the war - Snow is using her as a means to teach the rest of the Districts a harsh lesson, and Coin is using her to rally the Districts behind her.
This is a clash; one which needs to be resolved in some sort of climax.
Boggs the motivator
Boggs does two extremely important things:
- Makes Katniss realise that she herself is an important playing piece in this game
- Sets Katniss as squad 451 leader, confirming that she must go ahead with her mission (Kill Snow)
From Mockingjay:
"Sometime in the near future, this war will be resolved. A new leader will be chosen," says Boggs.
I roll my eyes. "Boggs, no one thinks I'm going to be the leader."
"No. They don't," he agrees. "But you'll throw support to someone. Would it be President Coin? Or someone else?"
"I don't know. I've never thought about it," I say.
"If your immediate answer isn't Coin, then you're a threat. You're the face of the rebellion. You may have more influence than any other single person," says Boggs. "Outwardly, the most you've ever done is tolerated her."
"So she'll kill me to shut me up." The minute I say the words, I know they're true.
-Suzanne Collins, The Hunger Games: Part Three 3 - Mockingjay, Chapter Nineteen. [emphasis mine]
Then, when he dies, he gives the "Holo" to Katniss, making her move further with her "mission":
"No. Boggs gave it to me," I say.
"Don't be ridiculous," she snaps. Of course, she thinks it's hers. She's second in command.
"It's true," says Homes. "He transferred the prime security clearance to her while he was dying. I saw it."
"Why would he do that?" demands Jackson.
Why indeed? My head's reeling from the ghastly events of the last five minutes--Boggs mutilated, dying, dead, Peeta's homicidal rage, Mitchell bloody and netted and swallowed by that foul black wave. I turn to Boggs, very badly needing him alive. Suddenly sure that he, and maybe he alone, is completely on my side. I think of his last orders....
"Don't trust them. Don't go back. Kill Peeta. Do what you came to do."
What did he mean? Don't trust who? The rebels? Coin? The people looking at me right now? I won't go back, but he must know I can't just fire a bullet through Peeta's head. Can I? Should I? Did Boggs guess that what I really came to do is desert and kill Snow on my own?
-Suzanne Collins, The Hunger Games: Part Three 3 - Mockingjay, Chapter Nineteen. [emphasis mine]
Seeing that it was a Rebel plan to set-off the double-exploding bombs (killing Prim), and not President Snow's idea [1] enables Katniss to realise that Coin is really just another substitute for Snow. Worse in fact that she will do whatever it takes, including kill children (including Prim) on public television to get her way[2]. Ultimately leading Katniss to Kill Coin instead of Snow!
This is the most important plot piece that she gained on her mission to the Capitol!
Peeta
Katniss sees and learns that throughout all of the tumult of the Squad 451 travels, Peeta does not kill her! He is capable of change and being changed. If she had not seen that for herself, she may not have been so open to him returning to District 12 and living with her.
She sees that through multiple occasions and opportunities for Peeta to kill her, he doesn't. She learns to treat him with a small degree of trust, and a huge amount of humanity.
TL;DR: Continuing into the Capitol was useful because:
- what happens there helps Katniss realize Coin is not to be trusted.
- Katniss was able to observe Peeta's recovery and begin to trust him again.
[Note: I have not seen the movie yet, so my answer will be solely book-based; I will confirm/rebuke my findings once I've seen the movieI have watched the movie, and confirmed that the events play-out nearly identically to the book.]