When it comes to running CNAs, we don't have to run them in a public cloud. The public cloud is the manifestation of automation and easiness so that we can use a ready-made platform with all the capabilities for end user consumption. You can compare public clouds to an airport where you can buy services to go from point A to point B without buying an airplane.
The term cloud-native is not related to the public cloud as we understand it. The public cloud is one of the vehicles that's used to deploy, manage, and run CNAs.