Wrong. The money gets distributed from two sources: the state, and the county via property taxes. The size of the district has no impact on the money-distribution formula for either.

If a county, or a state, wants to send money to poorer areas, they can do it without regard to the organization of the school districts. Often, the formula is targeted at the student: if a school has X low-income students, then they get Y dollars.

Edit: my comment pertains to large school districts, aka city-wide or within large counties. The suburbs get organized differently.