Importing nodes with Node export is creating them in "unpublished" status
At admin/config/content/node_export there is Reset values on import > [your node type] > Published checkbox, which is not checked by default. Checking this will reset the published status, which means nodes will be unpublished (that's how I interpret that anyway). So leaving it unchecked will not change the status. There seems to be no way to force nodes to be published.

If needed, on the new site nodes can be mass published at admin/content.