Now while it's true that your conflict can be anything at all, the trick is ensure that there's a reason for your main character (or any character for that matter) to respond to it. If it's a small conflict that suits a sub-plot then, no the fate of the world doesn't have to rest upon it, but it has to matter to someone otherwise it isn't worth exploring. As the centre of this conflict grows in importance to your plot, or is the basis of your plot, the more it is necessary for your characters to be very clear about what's at stake if this conflict isn't resolved.
As usual, I'll give you an example from a work I critiqued.
The Faerie World
In this story, a huge world-altering war is brewing between two sides of the Fae, and only the chosen ones will have power enough to stop the darkness.
Conflict? Yes. The world being overcome by darkness is a pretty big threat.
Does it matter to the Main Character? In theory.
This is where I have to intervene. The MC must act because the fate of the world is at stake, however, as a reader, I need to know the details of what's at stake if she doesn't act or doesn't succeed.
Darkness sounds bad, but what does a world ruled by it really look like? I mean, I can make my guesses, but I need to see some signs of what this means other than "it's evil". Unless of course we're headed for a double-agent type of deal where the Light ends up being the manipulative evil, and the Dark just gets a bad rep, in which case, yes, you want to mask the real evil a little longer than you would otherwise.
But regardless of how big and bad the evil may be, you need to ask yourself as a writer, "Why her? Why now?" And just replying with, "She's chosen!" isn't really motivation enough for me to invest in this journey.
Let's look at other famous chosen ones:
1) HP, the boy who lived.
Is letting Voldemort win a bad thing for both the wizard and muggle world? Yes!
Are we given the details as to why? Yes, because he came to power before hand.
Is Harry Potter going up against him purely because it's the right thing to do? No.
I've got your attention now, don't I?
Harry Potter, while a good person who wants to do the right thing would want to help out defeating Voldemort because he should, would probably not have become the leader that he was if he didn't have a personal reason to get involved.
Voldemort killed his parents - very personal. Out of the hundreds of random killings, this one means the most to our hero.
Voldemort specifically wants Harry Potter dead more than any other being alive - very personal.
As Harry Potter becomes more invested in life in the wizarding world, Voldemort up's the ante by killing people Harry Potter cares about - Sirius Black, Dumbledor, Hedwig, and of course the threat of losing everyone becomes more of an issue.
If Voldemort went on a killing spree in the first book, yes it would be unfortunate, but Harry wouldn't have had the time to solidify those bonds. Even the beautiful Hedwig would have just been an unfortunate loss as opposed to the heart-wrenching shock that it was. But these losses, and threat of loss, matter because we know how emotionally tied our Main Character is.
In my critiqued example, there is no tie. It's just an important adventure. So the note I left this writer was "Show me what she has to lose!"
Think about Frodo from The Lord of the Rings. Really he's just a hobbit who's in the wrong place at the wrong time, with a present that he shouldn't be in possession of. Now there's two things going for Frodo to drive him into action against Sauron and the One Ring:
1) He has seen first-hand how the ring has changed his uncle. No, Bilbo isn't evil, but there is the sense that something isn't right. Though this becomes more apparent after the fact, it's still something for Frodo to hang on to - a first-hand account of how people he loves can be overwhelmed by darkness.
2) The immediate threat. Evil is coming, it is coming for you, and it will kill everyone in its path! This is what gets Frodo out of the door! Self-preservation, and the protection of his people (even if they are ungrateful).
And while saving the world is an important part to the reader for understanding just how evil Sauron is, the lives of millions of people aren't the driving hero factor, they are more like a happy consequence to saving those who matter. It's an unfortunate human reality - we do care more for the one than for the many, despite what Spock may have taught us.
Now this isn't all to say that you can't have a hero who genuinely wants to save the world without any personal gain - but we still need to know why? Is our hero trying to atone for something? Do they have a loved mentor who is grooming them for good? Is it an ingrained personality trait?
I have to make a comment on this last one. No joke, I taught a student whose OCD made them only tell the truth, now it wasn't Liar Liar style where he was physically incapable of saying that the pen is red, it was more of a practical thing where inventing a lie just served no practical purpose and so he always focused on his real tasks/responsibilities. Really creative kid, and a great storyteller, but those inventions served a practical purpose. Lying about his homework didn't achieve anything, so he didn't do it. I mention this story because why not have a hero who must compulsively fix things, or make things good? At least we'd know what's in it for them, what their motivation is!
I know that these examples have been catering to the fantasy world, but remember that the same applies to your contemporary genre as well.
We can accept that a doctor needs to cure a rare virus because that's what doctors get paid to do, but we readers would be so much more invested if her sister, child, favourite grandfather had or died from this virus.
We can cheer on our corporate climber knowing that he won't feel fulfilled until he gets that big promotion and can finally make his father/wife/potential lover proud.
We can cry tears of joy for the quiet girl in class who finally takes centre stage and dance, overcoming the years of poor self-esteem that she has had to battle through.
All of these heroes can take us on amazing journeys that keep our hearts in knots, but only if we get a chance to invest our hearts to them. It's them we need to be attached to, we need to understand what they fear, what they dream about, who they care about, otherwise a trip to Mordor may as well be a trip to the supermarket - when what we need is a trip to the supermarket to feel like a mission to Mordor!