Mixed pile under the tree when I was growing up. One person reads the labels and hands them out, and then they're mostly opened one at a time, but sometimes you get two or three going, if someone is a bit slower or if there are young chidlren who are a bit more impatient.
order is all a bit random, as some people get more gifts than others, depending on how things are wrapped etc.
Doesn't take that long to write out - each person gets one from parents, one from santa, one from each sibling, and possibly a couple of extras from things like 'the elves' or whatever.
I think it focuses the children on the 'giving' aspect more, as you are looking forward to watching someone open something that you've specially chosen for them. It doesn't have to be a big dramatic all-eyes-on-them thing, as you can mix it up with eating/drinking bits for breakfast, looking at your own presents, opening a couple of things at once etc, if someone is really bothered by the attention aspect of it, and you can go fairly speedily through them if you want, but it can help children know that the 'giving' aspect of giving someone else pleasure is just as much fun as the 'getting' aspect of their own presents.
And it stretches the time out, makes it less of a frenzy that just bags of presents to open all at once, which I think is quite nice.