Filezilla is just a FTP client and has nothing to do with deployment. You manually have to go through all files that need to be uploaded, wich is a tough job.
There are other deployment tools that are much easier. For example MEAT (http s://getmeat.io), the online version is free for 3 project. And the offline version is free for Unlimited projects.
What do you mean with 'Filezilla is missing the public_html folder'? Arent't you already in that folder..?