All the land around COTA is Austin ETJ rather than full purpose. My understanding is under state law, a municipality has no right or responsibility over road construction or maintenance in their ETJ besides being a rubber stamp on the local community coming together and voting to form a Municipal Utility District (MUD), which would have the power of taxation to develop local road infrastructure. Otherwise, the only entities with any power to do anything here would be Travis County or the State of Texas.
If that community wanted to be annexed into Austin, I'm certain Austin would develop a plan for the roads there, but I doubt they'd agree to annexation