Hello,
the test you did with Swagger UI directly from the API reference doesn't work because it points to the wrong domain, that is https://www.thenewsletterplugin.com. To test API in localhost without HTTPS you need to set define ('NEWSLETTER_REST_ALLOW_NON_HTTPS_REQUEST', true);
in wp-config.php. Here you can find the updated documentation.
Kind regards,
Gioacchino