I think another issue we're having, is what is the definition of conflict? If we're discussing conflict in the context of drama, or narrative, in general, the term does not hold the same definition as it does in other contexts. There's our internal conflict as well.
But let's move away from the term conflict, for a while. Let's go back to our ancient Greeks, who seem to have meddled somewhat in this discussion already, and borrow their term for the concept: agon. This is the word that gives us protagonist, antagonist, and agonising. Its meaning approximates contest, which is less strong than conflict.
The main function of agon, is to provide a certain tension, to drive the story, regardless of the cause of that tension. It may be your classic melodramatic good guy and bad guy, it might be one man's fight against the sea (not to mention a marvellous marlin), or someone's growth over time.
"Then the Queen died of a broken heart" contains this tension, even if it is not explicit. There is the tension between the instinct of survival and the emotions of grief. Hemingway's shortest story has the same kind of tension, in its implicit context that the child never got to wear them. It's less poignant, to me, in the estate sale example. But well, in six words or less, your tension practically must be implicit.
Even in the most basic narrative, there is the tension of time - of what was, and is, and will or will not be - or conversely, like in De avonden the sense that nothing changes despite that being the normal order of things. Granted, that will not always be the most interesting of narratives, but that in itself stresses the importance of that tension.
It might be that I've spent too much time reading my scholastics, but a proper definition of the question, and the terms used, can sometimes enlighten things to a surprising degree. So, I guess that this is what, in a narrative context, I understand by the term conflict. And I hope it helps enlighten why I do think it is quite essential for most narratives - poetry is another thing, though, where the same rules do not always apply.