Synopsis
Given that MS/Zenimax seems to be opening up a bit more to player options, I'd like to suggest a dedicated story mode.

Justification
If you have not been playing religiously since day one, then you are constantly interacting with NPCs in the stories that talk about how you did this or that while here or there and mentioned various story points even if you have not done any of those stories or met any of those NPCs yet. It's incredibly aggravating, especially when there is no in-game guide to following the quests and visiting the places in the order that they were originally made for.

Target Audience
Players who wish for a more catered and chronological approach to the story with a smaller group of friends (Guild), being restricted to lore-friendly means of travel and discovery while hiding the mass of other players around them that take away from the experience.

Key Points

• When you first create a character, you can choose to create it in story mode. Once selected, the choice is permanent for that character.
• The character must adhere to the alliance race requirements, even if their account has unlocked the option otherwise.
• The character will begin in the original Cold Harbor prologue.
• Story Mode characters will only ever be able to see and accept quests from NPCs, objects, etc that they have unlocked so far in the story.
• The character will only ever see players within the same guild or group as the player. (in story mode, you are the hero, not the hundreds of people sprinting all around you)
• Unlocks happen mostly by zone. For example, Cold Harbor -> Khenarthi's Roost -> Auridon -> etc
• Once a zone is unlocked, then previously acquired account-wide things associated with that zone will also be unlocked.
• Wayshrines only function within a zone itself, and never between zones. You must use normal means, such as portals, ships, NPC captains, etc and they only work to unlocked zones.
• You cannot teleport to Houses; you must use the front door. (portals within homes still function, if zone unlocked)
• Game mechanic unlocks, such as jewelry, scrying/excavating, Tales of Tribute, etc will be adjusted so they can be unlocked and used within unlocked zones without needed to travel all around.
• The RNG quests will limit areas to those that are unlocked (such as Undaunted).
• PvP will be segregated for those in Story Mode and a less teleport-y method will be devised to get to Cyrodiil.
• PvP game mechanics and content will also be locked behind progression to how it would make sense story-wise with NPC introductions, character accomplishments, etc. For example, a commander would not trust you with an operation unless you had proven yourself during the course of your questing or previous lesser operations.
• There will be quest lines to include zone-based content, such as World Bosses, World Events, Locales, Set Stations, etc through various means (Fighters Guild, Mages Guild, Crafting Guilds, etc). A happy medium should be sought between instigated content through quests and independent exploration (maybe reward proactive players vs those following a map icon).

Crafting *A side note
To be blunt, crafting just needs a massive overhaul. In its current form, it's basically useless except end-game content to grind out a specific set you might want. In an ideal world, crafting would progress at about the same pace as a character does (blacksmith + cloth or leather + enchant + alchemy + food) and during their normal questing would acquire almost enough components to craft a full set of armor/weapon once every 5-10 levels. Personally, I think character leveling is just way too fast, but this is all discussion for a different topic. Should crafting be overhauled, in Game Mode, the various crafts would be fleshed out into their own stories with guild-like content for progression: fetch components, teaching, hunting down recipes, full set crafting requests by important NPCs you've met and know, etc.

Events
Since the story is locked into chronology, this opens the possibility to allow characters to experience the one-time events that took place across all the DLC releases. Perhaps a separate version of zones can exist pre- and post- event so every player can experience that. Some of them had content that required extremely large groups, so would need to be looked at. This also would open the possibility for players who are in Game Mode and under such heavy restrictions to earn previously offered things from the time of those events as well, though I believe the criteria should be altered to make story sense. For example, the Grand Psijic Villa could be a reward for completing the entire Psijic Order story plus some additional things (perhaps Antiquities excavation).


Pre-Emptive Responses
ESO is an MMO; hiding other players and only showing friends shouldn't be allowed; you should just go play some other small co-op game.
-> ESO happens to be the only Elder Scrolls game with multiplayer; besides, Story Mode would not be forced.
-> Personally, I don't mind the people, but I'm heavily put-off by the high-fantasy frills of all the unicorn vomit in ESO. I understand that having lava, ice, or moss skin while riding a burning ghost mechanical horse is fancy, but I don't like being forced to see it. (Maybe a graphics option that disables all that and turns all other player models into mundane soldiers on generic horses and all non-combat pets into cats, dogs, crows, and guar.)

There's no such thing as chronological, Zenimax made everything so you can play the main story and DLC whenever you want.
-> I can only assume that you've never actually read the NPC dialog; I've seen this argument so much on reddit, but it doesn't make any sense, sorry.

Dungeons are made to be independent of the story, there's no way to have them be chronological.
-> Yes, there is. Almost every dungeon has a theme based on the story content of the zone it is in. That theme has an introduction and usually a resolution, and so it can be made chronological. In fact, the experience for Game Mode could be amplified by integrating overland quests that incorporate the dungeons, delves, trials, etc.