Sorry, no, that is not how the SmartThings architecture works. It’s true that there is a Z wave controller inside the SmartThings hub, but SmartThings still requires it to check in with your cloud account for many activities.
At the present time, the only thing that can run when the Internet is down are “smart lighting” automations using the official feature that include only devices with device types that are eligible to run locally.
Basically all custom smartapps and all custom device type handlers and a number of official device type handlers are processed in the cloud. All routines are processed in the cloud. All the Smart apps listed in the marketplace in the official mobile app are processed in the cloud.
In addition, the official smartthings mobile app uses the cloud to talk to the hub, not your local Wi-Fi. So if your Internet is down, you cannot use the mobile app at all.
So if you have a motion sensor Which uses a stock device type handler and you use a smart lighting automation to have it turn on A zwave switch which uses a stock device handler, that will probably work if your Internet is down.
But if you set up the same functionality with the same two devices using a routine, that will not work when the Internet is down.
Also, none of the ST integration for Devices connected by LAN will work when the smartthings cloud is not available. So you will have no control from smartthings over your Phillips Hue bridge, for example, if the Internet is not available, even through a smart lighting automation.
Technically, all of that could happen, it’s just not the way SmartThings is designed. It’s primarily a cloud-based system.