The part of route 130 you are referring to is technically owned by texas, but under a long term lease/manage situation, I believe the lease is 50 years. It's private in practice. There are many more such situations in the US than folks realize. Chicago Skyway, Indiana toll road and others. Cities/states will contort structures to be able to say "we still own it!" while economically not owning it.

So, the ones in states that can't afford to build or maintain them are often private roads or PPP.

Australia is one of the top toll roading countries. Sydney had most tolled roads world record for awhile not sure who holds it now. I don’t know whether it is the state or private companies who operate them.

Since you live in the area, and are talking about toll roads, I assume you also probably know of the name Cintra, who are Spainiards. And Cintra is, in fact, a private toll road operator who has owned roads around Austin previously