Given the application-specific details of every application package, like Drupal or Wordpress or SMF or phpBB, etc, I don’t know how DH could possibly create a single tool like this. To move a site we just need to FTP down from one location and FTP back up to another (preferably with a tar before/after) … or SSH into one of the systems and the FTP to/from the other in one step.
After that, config files need to be changed for the database (which similarly needs to be saved and restored), email addresses, and the application’s internal references to its new domain.
It sounds like you’re trying to develop a site and then just pass it on to a client. It’s “sort of but not too” unusual that a client who has the ability to do this on their own would commission someone else to develop a site for them anyway. Since most clients need this service, consider a model where your service includes installation and configuration of the final site to the client’s servers, and then you can back off until they need you again.
Another option, if you’re really into automation, is to use Ansible, with scripts that you write move files, save/restore databases, and modify config files. This would be your tool, specific to applications that you write, that you write and then use everywhere. Again, consider the difficulty (OK, I’ll say ridiculous implausibility) of DH attempting to do this with a general purpose tool that is intended to serve every possible application and configuration need.