"providing them with SFTP access, or using a cloud storage solution would be better ..."
Well I'm not sure. Imagine this: a window with just one button and just one text area on it.
Initial state: text area says "ready", and button says "click here to choose file to upload" which invokes the browser's inbuilt file picker
Second state: text area says "upload myfile.dat?", and button says "click here to start upload"
Third state: text area says "uploading myfile.dat", and button says "click here to cancel upload"
Fourth state: text area says "successfully uploaded myfile.dat", and button says "upload another file?"
How can that possibly be worse than another solution?