Well, first, we should probably think about the nature of the city. Trying not to be "a disaster caused the city to split into two" and "poor/rich" divide like we have, maybe it could be that the "Dystopian" part of the city is the outer perimeter of a larger whole. There's an almost circular fence coiling around a much more well-faring part, though it's mostly filled with normal people instead of super rich folk. Resources are scarce, so the ones who were lucky enough to be inside the walls are the ones taken care of.
I think that might be a cooler way to create a "Satellite-like" wasteland than just being like "Well, there's just a part of the city where the poor people live. Scarce resources seems kind of essential for a 5Ds style game where everyone is in a duel gang.
Being inside the walls means you're like a scientist or an engineer, tasked with keeping the infrastructure of the city going. Outside the walls is manual labor; They recycle scrap to useful materials, create the food which is funneled back into the walls, take care of the sewage system, craft general commodities...
Security distributes food at checkpoints, but they're heavily guarded and it's never a lot. Some people manage to smuggle stuff from underneath their noses, and the currency everyone trades in is cards. Money has no real value, so being a duelist is the only way to prove yourself. This is why gangs band together; If they want to settle territorial disputes, they'll need whatever duelists they can get. Whoever's in control of an area takes care of the people in it, too.
I like this idea, and I think it still leaves options open for certain kinds of characters. I've been fairly into the idea of a mechanic character, and while engineers and scientists would typically be in the center of the city, could make sense still that some people got kicked from the core or whatnot. The son of an engineer for a KaibaCorp (or similar business) who for one reason or another gets exiled. The father figure begins toiling his life away doing manual labor, teaches their child how duel runners worked and the like. The child grows up, learns not only the basics of engineering but also how to duel so he can join up with a gang and try to either get himself or his father back into the center of the city...
Rambling off a character idea aside, just looking from a story perspective I think it creates an environment that can allow for a diverse cast to come together just out of necessity, while also allowing for more jovial situations. Duels between members of the gang for something like a spare ration of food could keep players dueling players some of the time rather than just NPCs the whole way through. The setting also allows for a diversification of smaller group goals along the way, like trying to raid a food checkpoint when food supplies are running lower than usual to take care of the people in the gang's area.
Simply put, I think it works well.
The goal of our gang is simple: Get inside the walls by any means. What you do inside of it when that happens is up to you. Of course it's a daunting task, and it's a dream really anyone in the outer rim shares, but it's a simple enough goal as to why people would band together.
Simple end goal, and good for the same reasons you've already mentioned. If people wanted a more grand plot, too, something could happen once everyone is in the core of the city, and people have to unite for a different reason.
-Need to hear from people if we're throwing speed spells in or not. Getting some mixed messages right now, so a definitive answer either way would be nice.
I'll be honest, I've been doing some research into turbo duels to keep me occupied during this time of instability, and I think having speed spells and turbo duels with speed counters is a good idea. There's plenty of speed spells from the anime that are out there to use, and balancing out speed counter issues is something that could even be figured out on a case by case basis when it would come up in a duel. It just adds something cooler than people standing around and dueling in the streets in terms of writing, and also shakes up the mechanics enough so that people could have two slightly different styled decks depending on whether they are turbo dueling or dueling like every other chump on the street.
-Need more feedback on the idea I posted above. If anyone has anything to add to it, anything they don't like about it, or any other suggestion entirely they're free to post it, but a "this is good" is also fine because then I'll know I'm going in the right direction.
I mean, only suggestions and ideas I haven't listed out above are names for the city or different parts of the city. I kind of like the idea of the center of the city being called "The Core." Don't have good ideas for the outer city's name, and the only gang name suggestion I would have is "The Outriders."
To the Master Rule question, I think the only real answer is to play in Master Rule 5...
Based on the arguments made and the little bit I've learned in how the game has changed, you're making some good points.