Enhance does update the site URL with a search/replace from the old domain to the new. Sometimes when this doesn't work it's because WP_SITEURL and WP_HOME have been overridden in wp-config.php which takes precedence over the setting that Enhance sets via wp-cli (or any changes you make in wp-admin or via the Enhance WordPress toolkit).
Another possibility is that something else is breaking wp-cli. wp-cli loads at least some of the WordPress code so errors in WordPress often propagate up to wp-cli. Try updating whilst logged in to SSH as the website user on the destination website, are there any errors?
wp option set siteurl https://your-new-domain.com --path=public_html
