Sorry, this is not possible because web browsers don't provide local path information to the server when uploading files, for security reasons.
The shinyFiles package lets you browse files on the server via a web interface, which is completely different. When you run an app locally with shinyFiles, the client browser and the server are the same machine, but the mechanism is still completely separate: for drag and drop (and file inputs in general), it is the web browser that is interacts with the filesystem, but for shinyFiles, it is the server is does it.