I could see a few different options. (I'm not sure if this is a spoiler since it's covered in the first few minutes of the demo, but I'll spoiler-tag it anyway)
1) they go for the happy ending because they don't want to make the most depressing Zelda game ever, since the canon result would end with Zelda sealed away, at least four or five playable characters dying, and Link being almost fatally wounded and taken away to heal as characters like Impa, Purah, and Robbie can only watch and mourn those who are lost
2) it turns out that they can't stop the Calamity from happening, but the Egg Guardian survives and travels around Hyrule for the next hundred years, bringing more towers out of the ground. These towers help Link with beating Calamity Ganon in Breath of the Wild. (I can't remember if the towers were there in Breath of the Wild's flashbacks, but if they weren't, this could be an elaborate way of explaining how they got there)
3) (this one seems unlikely, since it might be too complicated for a Warriors game) the whole game turns out to be a stable time loop - the Egg Guardian's presence causes the Calamity to happen. Not knowing when it was created, they send the Egg Guardian back in time however many centuries to stop itself, leading it to becoming the Guardian that tried to destroy the Egg Guardian before it went through the time portal
I'm sure there are plenty of other options; those were just the first three ending ideas that I had.