Local Connect only works directly with WP Engine and Flywheel but you can still get your sites in and out of Local using other methods.
If you’d like to load a site into Local it’s as easy as downloading a backup and direct importing that zip file into the Local app. Here is a help doc with some details around importing/exporting with Local:
As an alternative to being able to push up to your host you can utilize alternatives like SSH, SFTP or a migration plugin to be able to get your changes up to the live site.
If you have any other questions or concerns don’t hesitate to reach back out. We are always happy to help!