It does not matter where that money comes from. Core means that this is a ministry that the church no longer feels it can do without, just like worship. Some do, some don't. For accounting purposes it may be better to have one central account that everyone deposits to and withdraws from based on how much money each has. Most churches at least have a basic budget that covers mortgage, salaries, benevolence, utilities, etc.
Some churches have a budget that is made up of all of the budgets of the small groups.
But that really isn't the real question. The real question is why and how does a church raise money. If you raise money based on a budget, then you have set a limit to what you can raise. Solid churches develop tithers instead of raising money. I would not be surprised if the day comes when many churches may not have budgets at all and if they do, they will track human assests as much as financial.