No, you cannot use download_data to backup saved full-text indexes.
They are similar to data warehouses functionally, and they are stored similarly. If you want to “archive” data, you can programmatically extract it and copy it to another search index, and then save it. Or just pull it out and save it to cloud storage or something like that.
Thus, there is no single team solution for backing up full-text index data; you must write code to extract, write, and save them somewhere.