For SharePoint website, is there a feature that, developer can try all the settings in the Testing Site, and after the trials have been confirmed, the developer can click a Go Live button, then the changes will be applied to the actual Public Site?
No - there is nothing like this out of the box.
You can develop something like this depending on what you need to push. If the environments are the same, you can rely on the export / import methods for merging content, but that wont activate features for you.
If content is the same and there is no threat to overwriting production, you can move the entire content database from staging to production - and that would include features that are activated or deactivated.