Actually...not. Take a look at the Orange County Property Appraiser, or even the RCID website itself. The main roads are public roads, maintained and built by the District to benefit the business and residents of the District. That Disney is pretty much the only business/resident of the District doesn't matter, the District owns the land that the main roads sit on, not WDP&R.
EDIT: Just to be clear, I'm talking entirely about your first and second sentence. Your last sentence is basically correct. They go through RCID the same/similar way a private community goes through a county or city to have road improvements done within their community (see Happy Trails in Osceola County for an example). That doesn't mean the county owns that road through the private community, or the land it sits on. Now, the District/Disney relationship is a little different, but a similar idea is in place. For main road ownership (i.e. before the toll plazas and resort parcel lines), think of it like typical city/county and land owner relationships. The city/county owns the road and the land on either side of it for ROW, the land owners own everything on either side of the ROW. WDP&R owns the land on either side of the main roads and adjacent ROW, not the actual road and ROW itself.