> this is called the natural order of things
The natural order of things is only important inasmuch as it is a stable, desirable order. Lots of things we don't like happen in nature: starvation, disease, rape, murder, torture, isolation, immense pain. If these things happen to us we don't blithely say, "Ah, the natural order of things. It's all good!"
People want to preserve the natural order because they like the things they find in it. These things make them happy. Also, they like security. The natural order has persisted, with occasional mass extinctions and other disasters, for quite some time, so splitting off from the natural order into something that holds no promise of stability and security doesn't seem like a good choice.
But to the extent possible people who value nature and stability of existing ecosystems are still picking and choosing for themselves from what nature offers. No disease, please. Delayed senescence, please. No freezing to death or drowning. Less tooth decay.
And I suspect you don't adopt a throw-up-my-hands-and-let-shit-happen philosophy in your own life. You look both ways before you cross the street. You don't eat every mushroom you see. You stop walking when you come to a cliff edge.
So basically "[being happy to be alive and reasonably healthy] might be at the expense of other people, living or yet to be born. Not that we've done anything wrong – just that the circumstances that make this possible for us might preclude it for others" is just applying this same foresight and judgment more broadly. The qualitative difference is that it is not selfish. It is the difference between "I shouldn't step on this landmine" and "I shouldn't place a landmine where someone else will step". Why do this? Again, the same old reason: other people's happiness makes you happy; other people's sadness makes you sad.
When we want the natural order, it's only by accident.