Non-linear means that all those sidequests aren't necessary. You don't have to go through the quests in the order that they're presented to reach the ending, you can just skip them. See? Non-linear. It isn't do A so you can do B so you can do C so you will finally reach Z. That's just you misunderstanding the jargon, like you said.
And the doom isn't coming immediately, it takes months/years and most of the place doesn't even believe in it. All the sidequests you're doing, they're to get allies and weaponry so that when the doom does come, you'll be properly equipped and have armies on your side instead of just charging off on your own against it. And so you'll get a chance to talk to all those people and convince them to help you, of course! You aren't wasting precious time you could be saving to avert a crisis, you're building up good will so that when the crisis comes, you'll have reinforcement. XD