So this started out somewhat organized and then became a rambling mess. Sorry about that, I won't blame you if you skip over this reply.
Overall, I envision the playstyle kind of like Zelda games - 3rd person view, quests to fulfill, dungeons to clear, items to collect and carry, occasional monsters to fight along the way, and you learn about the regions of the world as you go through the game. All the while, you're telling an interesting story about the world. The entire game 'package' would be divided into separate campaigns, each one focused around the goals of a single faction. You could work on these individually.
Factions - So you pick a faction and then you do the campaign for that faction. You'd take the role of someone likely to travel, like an Ionian scout, Noxian spy, Freljordi hunter, whatever. Your mission is something kinda big for that faction, like find some text written by Averosa in the Freljord campaign, or learn about the Noxus war effort for the Ionian campaign. The path to complete each campaign would be designed to take you around Runeterra. So basically, you'd have 8 (or however many factions we have now) smaller games the player would get to work through. This would allow you to release different campaigns as you go, and as Riot updates/changes the lore. Also, it means the player would experience each faction differently. Like, the Noxus mission in Shurima would be totally different from the Piltover mission in Shurmia. Perhaps the same map, but different encounters, objectives, and learn different things about how those factions interact.
Solo or Team Play - Overall, I envision this as something you'd work through on your own. However, it would be cool if you could work with other people on particular missions. Say you are an Ionian who has a mission in Zaun. Well, you could find a Zaunite player willing to help you, maybe for a share of treasure. Or, a Noxian, Demacian and Freljordi could work together to overcome some big monster they found in the Shurima desert they can't face individually. Maybe these are part of the main storyline, maybe side quests. This wouldn't be like a typical MMORPG where you can see other people running around fulfilling missions. You'd have your personal question log and then try to find other people on the boards here to partner with to face the more difficult challenges (or maybe you somehow pull it off on your own and get more xp).
Champions - I feel you almost have to include Champions somehow because they have been such an integral part of what Runeterra is - like, if I can't watch Gragas and Braum have a drinking contest, why am I even here? These would primarily be NPCs giving missions, or even just amusing dialogue. Still, they could take part in some missions - like helping Talon sneak into a Demacian outpost in exchange for information, or following Taliyah as she guides you through shifting sands in the desert. Champions could also be 'dangers' you face in dungeons, like avoiding Ahri's charm in the Ionia forests.
Official Lore - I understand your concern about wanting this to integrate seamlessly with existing Lore, but I think anything is up for change with Riot in the long run. Honestly, stick to the lore as much as you can, but don't be afraid to be creative about filling in the missing pieces. You're going to need to make the call in a lot of instances on everything from appearance and style to cultural interaction and geography. And that's ok! I'm sure Riot would love to see your creativity there, even if it's not 100% in sync with their plans.
Note, I have some experience making maps, so if you ever need a dungeon map or anything like that, I'd be happy to help. Also, I like to pretend I know how to write creatively, if you need help with plot or storyline or things like that.
Edit - Also, each faction you pick could have a different combat specialty. Freljord archer, Demacian fighter, Ionian mage, whatever. You'd still get through your campaign ok, but it would make the instance of cooperation and team fight more interesting. Also, being able to aesthetically personalize your character, like in Borderlands, would be cool.
's Glorious Evolution would go to the Freljiord where the war between tribes takes place, or how 
would journey to Targon where fellow sun praisers like 
would aid him in bringing back Shurima to its sun praising glory.
to have an updated lore just like 
did. I want to explore his times with the Screaming Yipsnakes, how he fought in wars, or how flying a helicopter qualifies for fighting against champions like 
and 
who both fight with only swords and axes.
