SmartThings isn't (Smart, that is) anymore. How does one install an existing Smart App for a device in new app?

Do you mean smartapp or device type handler?

If it’s a custom smart app, you can install it by using the old classic mobile app and then it will run even when you are using the new connect app.

If it was a custom device type handler, it may need modifications before it can be used with the new app.