Order by link weight

Weight is included in the output, so you can use it to order. See for example:

       "link": {
            "id": "menu_link_content:32bd5d0f-0c96-470c-bb17-f30bfd113638",
            "weight": "11",
            "title": "phanecric",
            "description": "Description of phanecric.",
            "menu_name": "main",
            "provider": "menu_link_content",
            "parent": "",
            "enabled": true,
            "expanded": false,

Feel free to re-open the issue if this does not serve your needs.