##On the contrary, majority of the Fellowship did feel that destroying the Ring was important
Being the Ringbearer, it would be self-explanatory that he would definitely want the Ring destroyed, to relieve him of that burden. This very quote summarises his willingness to destroy the Ring.
'I will take the Ring,' he said, 'though I do not know the way.'
Sam's choice in going was more of protecting Frodo, while he went to destroy the Ring. Sam is utterly loyal to Frodo and would go the the ends of the (Middle) Earth with him.
'But you won't send him off alone surely, Master?' cried Sam, unable to contain himself any longer, and jumping up from the corner where he had been quietly sitting on the floor.
'No indeed!' said Elrond, turning towards him with a smile. 'You at least shall go with him. It is hardly possible to separate you from him, even when he is summoned to a secret council and you are not.'
- Merry and Pippin
Merry and Pippin had similar reasons as Sam in going, but it was more of a great friendship between them. Don't forget that they too were willing to start of the quest with Frodo in the beginning, and are now equally willing to go on because of their friendship and loyalty to him.
'It's most unfair,' said Pippin. 'Instead of throwing him out, and clapping him in chains, Elrond goes and rewards him for his cheek!'
'Rewards!' said Frodo. 'I can't imagine a more severe punishment. You are not thinking what you are saying: condemned to go on this hopeless journey, a reward? Yesterday I dreamed that my task was done, and I could rest here, a long while, perhaps for good.'
'I don't wonder,' said Merry, 'and I wish you could. But we are envying Sam, not you. If you have to go, then it will be a punishment for any of us to be left behind, even in Rivendell. We have come a long way with you and been through some stiff times. We want to go on.'
Now for the rest of the Fellowship. Aside from Gandalf and Aragorn, Boromir, Legolas and Gimli didn't really know the hobbits much and their intentions of going with him may not be out of friendship. However they are certainly willing to go with him, as Elrond surmises:
'And I will choose you companions to go with you, as far as they will or fortune allows. The number must be few, since your hope is in speed and secrecy. Had I a host of Elves in armour of the Elder Days, it would avail little, save to arouse the power of Mordor.
Gandalf's intentions are as clear as Frodo: Destroy the Ring and save the world. It is his duty after all for coming to Middle-earth in the first place - the defeat of Sauron.
'The Company of the Ring shall be Nine; and the Nine Walkers shall be set against the Nine Riders that are evil. With you and your faithful servant, Gandalf will go; for this shall be his great task, and maybe the end of his labours.
- Legolas and Gimli
Consider what Gimli says in this conversation:
At that moment Elrond came out with Gandalf, and he called the Company to him. 'This is my last word,' he said in a low voice. 'The Ring-bearer is setting out on the Quest of Mount Doom. On him alone is any charge laid: neither to cast away the Ring, nor to deliver it to any servant of the Enemy nor indeed to let any handle it, save members of the Company and the Council, and only then in gravest need. The others go with him as free companions, to help him on his way. You may tarry, or come back, or turn aside into other paths, as chance allows. The further you go, the less easy will it be to withdraw; yet no oath or bond is laid on you to go further than you will. For you do not yet know the strength of your hearts, and you cannot foresee what each may meet upon the road.'
'Faithless is he that says farewell when the road darkens,' said Gimli.
I think, although both Gimli and Legolas said that they would only accompany the Fellowship as far as the Mountains, in their hearts they were truly willing to go on. And they did.
Don't forget about the dislike they had for each other at the start. That is another reason why they'd continue going on: one didn't want to be seen as weak by the other.
- Aragorn and Boromir
Aragorn and Boromir are certainly trickier to explain. As your quote suggests, Aragorn is going to Minas Tirith with Boromir.
'I would have begged you to come,' said Frodo, 'only I thought you were going to Minas Tirith with Boromir.'
'I am,' said Aragorn. 'And the Sword-that-was-Broken shall be reforged ere I set out to war. But your road and our road lie together for many hundreds of miles. Therefore Boromir will also be in the Company. He is a valiant man.'
Boromir sole purpose of coming to Rivendell was to answer a riddle, and presumably also to gain powerful allies in the fight against Sauron. He's the heir to the Steward of Gondor, the most powerful (good) military nation in the South, and would feel that defending his city is in his best interest. In other words, unlike Frodo and Gandalf, he cannot see the bigger picture beyond the safety of Gondor. That is the reason why he succumbed to the Ring, he needed that power to help Gondor, and was not willing to bring the Ring to Sauron's domain itself.
"You will take the Ring to Sauron and sell us all!"
Aragorn, being the de facto King of Gondor, knew that should Gondor fall, the Orcs would have free rein into the Northern kingdoms. But he could see the bigger picture, unlike Boromir, and did change his mind later on.
##Bottom line; however...
The Fellowship's purpose was really just to help Frodo reach the end goal. There was no oaths taken promising to get to Mount Doom with Frodo, but rather help him get there. Why? The Ring was a temptation. It doesn't matter if 9 or a hundred people go with Frodo into Mordor, they would eventually succumb to the temptation of the Ring, which only gets stronger as they go on. Elrond knew this.
The Ring-bearer is setting out on the Quest of Mount Doom. On him alone is any charge laid: neither to cast away the Ring, nor to deliver it to any servant of the Enemy nor indeed to let any handle it, save members of the Company and the Council, and only then in gravest need. The others go with him as free companions, to help him on his way. You may tarry, or come back, or turn aside into other paths, as chance allows. The further you go, the less easy will it be to withdraw; yet no oath or bond is laid on you to go further than you will. For you do not yet know the strength of your hearts, and you cannot foresee what each may meet upon the road.
It is not a matter of unconcern for destroying the Ring, but rather concern that everyone gets corrupted while doing so. Fewer people tagging along means less risk of getting corrupted. Even Sam is tempted at the end, but his impressive willpower and common sense triumphs.