It definitely matters.

The story matters to me, it holds the entire game together. It doesn't have to be spectacular or revolutionary, but it still needs to exist. Otherwise, it's just a collection of minigames and quests - that's the main reason I'm not rebuying and replaying Saints Row 4 on PS4.

It really depends. As far as I'm concerned, "open world" isn't a genre, it's a description for games within other genres. In open world sandboxes for example, no, I don't think an impactful main story line is generally of importance. For many other open world games in other genres, though (like the previously stated FFXV/XCX), I think a main story is very relevant.

Perhaps I phrased it wrongly. From what I can see from reviews of Majora's Mask, the main story is get the 4 guardians, repel moon, kill the skull kid because reasons, you saved Termina horay. That's the main story. You can also beat the game without doing any of the sidequests (or so I hear). However once you partake of the sidequests, sure the main story influences it, but it's not the main story but rather a side story. It's like saying Fire Emblem's Support Conversations are the main story...but they're clearly not despite some of them being influenced by the main story of constant war and hardships. In fact the poll's options is "for cool side stuff" which I categorize Majora's Mask's sidequest stories and Fire Emblem's Support Conversations to.

If this continues, I won't reply any further because I feel like it'll delve into more opinionated debate just so you know.

Depends on the open-world game. Games with smaller worlds (like Assassin's Creed) are dense enough to not feel lost in it, which is why you can play it without forgetting the story. But games like Skyrim are way too huge and have too much stuff you can do. I just can't care for the story of it as there's so much stuff to distract the player.

Short answer: It's not that simple, but yes.

I think that a developer needs to give the player a sophisticated, disbelief-suspending, and unobtrusive reason to act in the world you're in, regardless of if the game is open world or not. Whether that equates to a simple story, a complex story, a deep story, a shallow story, or no story at all depends entirely on the context the game gives you.

If I'm playing Mario Bros., saving the princess does those three things.
If I'm playing Street Fighter, the mechanics do those three things.
If I am playing a Earthbound, its simple plot does those three things.

If I'm playing an open world game, and there's a clear attempt to introduce lore and world building of any kind, there needs to be a story there that matches the complexity of the lore being provided, and it needs to be compelling enough for me to want to play through it.

That's why the Elder Scrolls fail to me. They don't do those three things. The reasons Bethesda gives its players to traverse their worlds aren't sophisticated, they don't make me willing to suspend my disbelief, and they are completely obtrusive to the experience; if someone choses to play through the story of Skyrim, they are actively anchoring themselves to an inferior play experience that runs counterintuitive to the world and mechanics the Bethesda has provided you with.

The fact that Skyrim is synonymous with mods is proof of this. Again, this varies from game to game, but in order for a game like, for example, Skyrim to work under these qualifications, the game needs to provide the player with a way to want enjoy the plot without it clashing with the mechanics and playground provided to you.

This largely would suggest that the story be extremely none linear, and likely very simple, instead relying on well written, world building side quests to fill in the meat of the experience, not unlike something like Majora's Mask. Many open world and sand box games don't achieve this well, and suffer as a result, because your actions aren't framed well and don't feel meaningful.

I'm not gettng heated over this, just that I kinda don't understand what you mean and what the topic is.  Does the main story matter in open world games?  The main story is the quest for the 4 guardians.  Therefor, for this open world game, it does.  The story itself drives the side quests.  Its more a question of, are the side quests fun overall, rather than the story making the side quests fun.  I think I'm not finding the right words here.

In my opinion the Story/Setting/characters are the most important things in a singleplayer game. You need a world that feels interesting, characters you care about and a Story that gives you a deeper sense of why you Play the game or want to come back again the next day.

I really like the Assassins creed games because they let you Play games in historical cities and meet historical persons. They are great open world games because they have an interesting Story, great character interactions and a linear Story.

I really liked Red Dead Redemption because it has an interesting Setting (wild West), interesting characters and a rather linear Story.

I have some Problems with GTA... It is interesting in playing around big american cities, but the main Story is weak because it is lacking an overall quest or feeling of importance. You just try to get rich and most People you meet will betray you or die anyway. You can do tons of sidestuff, but most of that like Shooting the Pidgeons or doing all Monster Stunts is just repetitive.

My biggest Problem with Open World games was Oblivion... While it looked really great it was the most boring game that I ever played. The Main Quest was really weak.... and the killer Thing was that the main quest implied that the world was on the brink of a huge catastrophe but you could take all your time in the world to do hundreds of unimportant side quest or try to start a career in the different quests and ignore the global threat... What's the Point of becoming head of the guild when the world is about to be invaded? Thats pointless.

Oblivion was a game that failed to connect the Story with the game world. And that is what I have felt with many western RPGs/Action-RPGS. An uninspired main Story ... and then Keep the Player playing for 50 hours or more just with repetitve sidequest (kill x enemies, collect y items, etc....). Or gear farming by´killing the same enemies over and over again...

It really, really depends on which is better.

If the story is good, but the side quests are meh, I'll probably soldier on through the game and hope it gets better. This kind of stuff makes the gameplay look bad to me.
If the side quests are good, but the story is meh, I'll tune out on the story. Plain and simple. If the story is bad enough, I'll drop the game, but I have low standards.

Now, if there's a big disconnect between the gameplay and story...I'll drop it.

Hmmmm that's a tough one. The main story is not what makes the open world game great, but without it I do not think I would be as invested. In SKyrim's case, I played it for the side stuff