Actually, in several states the two-party system is baked pretty deeply into the state-level laws. Pennsylvania, for example, has a primary system where one has to declare whether one is Democrat, Republican, or Independent, and then must vote on the ballot corresponding to that declaration (with the Independent ballot holding only referendum questions and not any input into the primary process).

Those elections use the state's voting machines and vote security infrastructure. That's functionally equivalent to being state-funded, even if the funding is in terms of human-power and machine use donation and not cash.