In high school I would lure players to my games by offering them the chance to play litereally anything they could imagine (which inevitably led me to realise that a lot of people really aren't that imaginative...but that's another story). I had enough Rifts and Palladium sourcebooks that I could pretty much allow any appearance of character, and some quite powerful ability sets, purely from first level characters. I ran games with sentient machines (often non-humanoid), dragons, thinly disguised homages of popular super heroes, shapeshifters, devils, fey...all sorts of things. The trick lay in trying to develop ways that such disparate characters would end up working together, and trying to develop stories where everyone would have an opportunity to shine, and everybody would have times when their ability set just wasn't appropriate to the challenge being faced.
The fact that I'd allow literally any type of character drew people to my games. The fact that I tried to make the stories interesting for everyone, and gave everyone the opportunity to have good moments and bad, kept them at my games. It was gonzo, but it became epic...not due to over-the-top characters but due to meaningful interactions between those characters and the way they'd change the world (often to discover that they'd only changed one of many timelines, and a Rift would take them to a similar realm where those changes hadn't happened).
Over years, I realised that an "anything goes" approach to starting a campaign can be hard work, mostly hard to keep the coherency of the storyline, but also to keep finding reasons why such disparate characters would stay together. I think this is when my exposure to Vampire: the Masquerade and Werewolf: the Apocalypse hit. Other players at the time asked "Why would you want to play a game where everyone is stuck playing the same thing?", and I didn't have a good answer at the time beyond "Because they're cool"...which inevitably led to the World of Darkness crossover games with Werewolves and Vampires joining forces against the Technocracy and ghostly agents in the background moving everyone around like pawns. I heard numerous stories like this so it wasn't just my circle of friends who didn't seem to get the whole idea of focused campaigns.
Moving through social groups and gaming circles in the late 90s, I found a few players who really blew my mind with the gaming style. Not only did they restrict games to a single type of character, they'd focus on a specific subtype, then play within the conventions of that groups to develop diverse characters. Instead of Werewolves, they'd tell the specific tale of a pack of Silver Fangs trying to regain their heritage and strength to confront the horrors lurking in the shadows, while fending off political rivals...instead of Vampires, they'd weave narratives specifically focused on Setites of a specific generation as childer of a shadowy spymaster who had waged an eternal war against Egyptian foes who'd spread across the world. This meant that the groups didn't need to find ways to explain why they were together, and the story could really focus on other aspects such as "family" dynamics, unified fronts against superior foes, and exploration of the world beyond rather than constantly focusing on the tensions and bonds holding the characters together.
Both are valid types of story, but it's nice to have options rather than telling the same story over and over (despite having different settings and different characters).
Somewhere between the two extremes lies the idea of having a restricted range of characters. Perhaps playing D&D but making all the characters dwarves, maybe eliminating magic users from your game, or making all the character's lawful because they are members of the town guard (that's not to say they're all good, some might be corrupt and Lawful-Evil). A restriction sets a tone, I like to think of it less like a limitation, and more like setting an impetus for the direction of the stories I'd like to tell. It gets everyone on the same page from the very beginning. It's like an artist restricting their colour pallet to get a specific feel in their artwork.
A lot of recent games really play with this concept well. I'm thinking of Grey Ranks and Night Witches as examples, where a specific character set up implies a certain type of story, which in turn allows to you tell a more nuanced story because you've got all the shorthands and genre conventions within that setting to play with, you don't need to explain everything from scratch because there is a character present who doesn't really belong.
To take this back to my last post, and the thoughts I've been having with the LARP, I think that's one of the problems I've seen so far. The open character generation system has seen a wildly disparate group of loners and charactes who just don't particularly fit, and these players haven't been guided in such a way that they can integrate within existing structures in the game. The "Clans" who have developed specific limitations in the character types they'll allow have remained strong (even despite certain key players coming and going), but even then the clans are so diverse that it becomes hard as a coordinators to tie different groups together on a regular basis and maintain coherency in the overall plot.