I'll try to surprise you, but I do ask you bear with me, because I'm trying to be succinct in my explanations, and some of the theoretical scenarios I made up on the fly.
Cho'gath was a voluntary prisoner. Even so, Cho could be used by communicating to Malzahar and acting as the elder voice of the majority of the Voidlings, and could even be using his position to speak to other prisoners. One of my favorite theories was that while in prison, Cho'gath would strike a deal with Nocturne (Who can't really have any development outside of murder all magic users, unless you completely rewrite his story) which in turn would make Nocturne Void-tainted and create a ground for the fall of the Institute eventually. Cho'gath could theoretically be the inside job that brought the Institute low.
Without the Institute, unless he's killed ASAP, he's going to simply and in all honesty eat all of us and keep us imprisoned forever.
Fiddlesticks could be used as a way to herald a new champion, being the Harbinger of Doom, and be used as a plot device for the arrival of who his master is which could range from the Ruined King of the Shadow Isles to a completely new champion, but could still be in line with something that could be built up towards.
Morgana was in a stalemate with Kayle, her story about revenge is actually rather simple. She can wait for thousands of years, she's immortal, she doesn't have the same deadlines you or I do. With that said, Morgana's story could have been explored more upon by speaking to other champions, particularly other exiles who have radical thoughts, this is why I thought Morgana would have tea time with people like Riven, Diana and Annie to name a few and would be used as a way of compare/contrast with these other characters.
You may notice that I have the running idea of champions interacting with champions. This is what I think the Institute did very effectively as a background tool, to create a central hub which allowed champions to interact with one another so you can have Darius meet Garen (against his will) and be forced to not slaughter the chump (against his will) and be forced to interact with him diplomatically (against his will). You can argue, and be right, that Darius can meet Garen at a diplomatic meeting outside of the League, but what about say Nidalee meeting Ashe?
Nidalee is so far south she's never going to go to the Freljords on her own accord.
What about the Voodoo Lands and Annie, why would she ever possibly leave and interact with the world at large without sounding rather silly? Annie will travel all the way to the Freljords for no real reason because she has no reason to go there and freeze her butt off, she doesn't know who or what is there or anything of the sort.
Why would Azir say hi to the Ionians? They're of no importance to him and his empire, and they certainly don't interact with Shurima. With the League he'd be forced to see people like Irelia and Karma, and he'd suddenly feel like peacocking in front of the people who are considered by every other modern nation to be the best diplomats on Runeterra.
The League also provided a barrier, sometimes crippling admittedly like in Kalamanda (it could have been handled better the direction and diction itself), to stopping people from outright slaughtering each other.
I want to see how the Noxus vs Ionian war takes place now, because it makes no sense without the League.
The reason why Noxus could attack Ionia before is because the League was preventing Demacia, and thereby any other country/city-state from attacking one another who are in the League. That also meant Noxus couldn't fight Demacia anymore, so they had to turn their sights on to another country, which was Ionia, a trading partner of their's.
Noxus invaded Ionia and their initial invading force was decimated. They didn't like that a whole lot, so they sent their full force and asked Zaun to join them with their hextech machines of death, who were then joined by Bilgewater who saw an easy, and rich, country for plundering, and the League did nothing to stop it. This was a move done by the League to coax Ionia into joining them, that this is what happens when they're not protected. Demacia didn't attack Noxus because they could not, they were forced to be held back and Noxus could focus all of their efforts on trying to stomp out Ionia.
Without the League in place, why would Noxus ever dare go to war with anyone else but Demacia? The very moment they open themselves, Demacia will storm them and tear them to shreds. Lux's lore has her being a spy for Demacia infiltrating Noxus, she can't be the only one who has this job, so if Noxus goes to war against anyone but them you can bet that Demacia will know about it. The very moment Noxus leaves themselves exposed, Demacia will swoop in and slaughter them.
Perhaps Demacia is too weak to do so, which in that case, why doesn't Noxus just stomp out Demacia, their main enemy? The only reason Noxus looked to any other city-state/country is because they could not attack Demacia, they are not only in that stalemate but they could not break the shaky treaty the League created or else it'll end in a Pyrrhic victory for either nation. They looked towards Ionia because they saw them as weak in the military sense but would provide an abundance of resources and give them the advantage needed to expand more, develop faster and gain the money required for all of these expenses with Demacia being able to do jack and shit about it.
No one likes Noxus. They're dicks. They have a giant ass skull mountain in the middle of their city with a castle for crying out loud, Dr. Wily designed it or who knows. Noxus needs to be rewritten to be a little less over the top evil to create a better balance, because it's really hard to argue you're not that bad of a guy when you have lynched corpses swinging behind you and your only explanation is, "Sometimes it's good to be bad." If Noxus exposes themselves, then Demacia and anyone who doesn't like Noxus or them being allied to Zaun will obliterate them (Piltover taking out Zaun's main ally to reduce the overall economic strength of Zaun? Hell yes). But if Demacia is too weak to send such a fighting force, then why doesn't Noxus focus their efforts on Demacia? If Demacia is only on the defensive, that means you pick away at them, cut off their supplies and you choke them in their own city walls, it's a classic tactic and it is a minimalist approach when using your resources.
Noxus and Demacia have to be equal in power, they have to have that stalemate or else they're simply too close to one another and can conquer the other when the opportunity arises, this is what the core narrative of the League of Legends story was built around. Without the League, Noxus can't leave and interact with other city-states/countries nearly as easily because it may open them to Demacia's counter attack. But on the other hand, that's also been retconned, Noxus' Barbarian Pacification Campaign was taken out of Trynd's story I believe.
What the League did was promote interactivity between champions that could not interact normally or would not interact normally, and give them a chance to develop as characters and show off sides that wouldn't necessarily be seen. The question "What would Janna think of Morgana" is not farfetched if she meets Morg in the League, but is farfetched now because there's no reason for Janna to go to Noxus and meet Morgana face to face. I found it to be a nice touch that it was Janna who taught Karma how to use her powers in a more offensive way, because Karma joined the League and met with Janna and became friends with her, and saw Janna as a teacher.
The League was always meant to be a setting, a backdrop, the stage on which the characters acted upon. What it was meant to promote was character interaction, which is what a lot of the Journal of Justice did do (Trundle befriending Soraka as another example). Truth be told, the JoJ was not the best vehicle for narrative in a video game. I don't mind they took it away, but what I don't understand is that saying the JoJ, not only a written medium but a series of stories all collected and made available to you on a whim in the client, is less accessible and less "desirable" of a storytelling medium than the short stories being published now that disappear because they're on the forums.
Try looking for Azir's short story, Xerath's short story, Amumu's short story, Rek'sai's short story, Sion's event, on the main League website. It's obtuse to do so, it's actually a pain to do so. The Journal of Justice it was in the client itself, click one button you have lore.
Why did I bring up the JoJ? It was synonymous with the Institute of War, really. When it stopped being produced, the League stopped being mentioned in lore and now it too has been erased. Meanwhile the question I have is how was the JoJ a more inferior way of storytelling in comparison to the short stories of Azir, Xerath and Amumu? The JoJ was considered to be a failed form of narrative, which I think is true, since it was meant to be a gossip magazine rather than the main vehicle for narrative. And yet, I know more about champions via the JoJ and their motivations than I do from Ezreal's new 3 sentences that says the cocky explorer is easily embarrassed by praise.
What was it that the League really prevented, in the end? Meaningful consequence in a game where you are already eternally resurrected? If there was more meaning needed, then restructure, clarify, detract the power of the League, "nerf" the might of summoners because for some reason people think they're all uber powerful (despite Wukong and Talon mopping the floor with ye olde summoner, in comparison to one of the 3 triumvirate leaders of the entire League telling Swain to b.t.f.d), there were other streets one could have traversed to simplify the League and make it the setting it was intended to be, for all the characters, the people you play, to interact with one another and develop through interaction.