Story is important because as a DM part of your job is being a storyteller. That is a foundational competency of DMing.
At some tables, yes, but not at all.
Sometimes the DM is more of a story-wrangler, left trying to sort out the various stories and plots her players have cooked up and somehow find common ground between them so as to tie them together a bit.
Sometimes the DM doesn't bother: the game is a series of disconnected adventures that just happen to have the same PCs continuing from one to the next with no overarching plot or reason.
I don't feel I should have to explain that but if you want you can pick up the DMG and read it; it mentions story throughout the book. As an example, "It is good to be the Dungeon Master! Not only do you tell fantastic stories about heroes, villains, monsters and magic, but you also get to create the world in which these stories live. (5E DMG pg. 4 Introduction).
Here you're conflating setting construction and backstory (which I agree are very good things for a DM to know how to do) with in-play storytelling, which if done wrong can very quickly and easily lead to the players/PCs being railroaded through the DM's pre-made story.
I'm not sure if the 5e DMG cautions against railroading or not - it's been a while since I read it - but if it doesn't, it should, strongly.
It seems not only is there a misunderstanding of foundational DM competencies but also of analogy. I used Danny Devito because that is what I picture when I think of say a 2E character whose average abilities scores (if using the default 3d6 keep in order rolled method) would be in the 9-12 range. I picture somebody with physicality of Danny Devito picking up a sword and saying I am going to fight. It is the furthest thing from heroic fantasy or sword & sorcery that I can think of BUT it is very Game of Thrones like so I guess I can see why you like it. To break it down as simple as possible not since 2E is it assumed that the PCs are just average folks (ability wise) who just decided for whatever reason to risk the adventuring life. Instead it is assumed that whatever their socio-economic/personal origins that the PCs are above the average mien (ability wise) and are marked by destiny to have an impact on their world.
Yes, and as far as I'm concerned this is a flat-out error in design.
Why?
Because the 0-1-2e model covers more ground. You can start as a nobody and work your way up OR you can start as a hero and go on from there, via the trivially-simple means of where the DM sets the starting level of the PCs: 1st or 3rd or 5th or whatever.
4e-5e don't give you that flexibility. There's no mechanics that cover the gap between commoner and 1st-level character, thus you simply can't start as a 'nobody'; and in 4e in particular that gap is immense.
In short, it assumes the DM cares what happens.
Which I take to mean you think a 0-1-2e DM doesn't care what happens?
Statements like this makes it hard to take you seriously - which is a shame, as though I disagree with almost everything you say you otherwise generally say it well.
Again, I never said that OSR game can't be intense or enjoyable (again provided the group has buy-in). I would not like that style of play nor do I think that style of play would be good for the modern game but I never said some people don't enjoy it.
Not everyone is going to enjoy anything. But I do think the OSR style as default would be good for the modern game in that it's both easier and more pleasant to relax restrictions and-or make things easier than it is to impose restrictions and-or make things harder.
As far as the nostalgia part, yes I will stand by what I said on that part: many OSR people I have meet seem to look back on the game and not realize that they enjoyed those games to a large extent because of the house rules or rules being ignored.
Nothing wrong with that either. Remember, in general (and some Gygax admonishments notwithstanding) the over-riding ethos in 0-1-2e was that the rules were largely guidelines, to be amended as an individual DM saw fit. Yes this meant that every table played differently, but it also meant that the game could be made to suit more tables.
It was 3e that brought in the idea of rules-as-law. 4e kept it*, while 5e has specifically tried to return to a more 'rulings-not-rules' ethos.
* - yes there was the legendary Page 42, but in reality how many 4e DMs put that to much use?