I’m a big fan of making my own excerpts for pages, strictly due to producing cleaner search results pages. https://www.wpbeginner.com/plugins/add-excerpts-to-your-pages-in-wordpress/ You add a quick line to your child theme’s functions.php file and then you can add excerpts to pages.
Are you using Cornerstone as a standalone plugin, or with X Theme? If it’s standalone, your theme will largely dictate what appears in the search results, which might explain the shortcodes showing instead of text. Many will default to showing the excerpt if it exists, which makes the solution above work.
One plugin i run on almost every site is Relevanssi https://en-ca.wordpress.org/plugins/relevanssi/ which has the option to execute shortcodes. It also allows for more granular control over what post types get indexed for the WordPress search.