Many groups dabble with this in some forms. For example, here are some subjects of gameplay that lend themselves well to this: economics, politics, war, and maybe espionage. Notice that these are the types of things you do in board games a lot. But you could totally have something like construction, conservation projects, running a business but focusing on the non-financial parts (e.g. running an opera house and managing the actors, the playwright, the stage production, etc.), going through religious rituals and sacraments, forming relationships, and so on.
While plenty of people have made downtime mechanics, and plenty of others recognize the fun of this sort of thing, it’s very rare that I find games that actually equip the DM to run this. There have been resources and websites like Obsidian Portal or World Anvil that cover a lot of what I'm about to describe, but most groups don't realize how to take advantage of the potential. All too often, these are treated as resources for the DM rather than the party. So I'm sorry 5th Edition, it's not enough to create an inflexible minigame for every specific "downtime activity" that occurs to you. The only way you'll ever achieve a true player-driven sandbox campaign is by letting their imaginations drive the car and merely providing the fuel and tools to guide that.
So lets talk about the fuel and tools.















