It would be nice to be able to directly upload your site to your own hosting space within Bootstrap Studio.
Search the forums. It's been requested countless times and while it might be convenient, that there are free, third-party FTP programs out there, so I can only assume the devs feel it's not crucial to the use of BSS.
you can run a script on export. So that feature is already present. Write a small sftp/rsync/rclone script with your hosting detail and sync you local export directory with your remote hosting.