Just to update the answer for today:

As of now, Visual Studio App Center is scheduled for retirement on March 31, 2025. Maybe for anybody considering it: Dont'.

Here are some options you might want to consider:

As for CodePush. You can habe a Self-hosted CodePush Server:

https://github.com/microsoft/react-native-code-push https://github.com/htdcx/code-push-server-go