If all of our suggestions did not work, then that means the issue is more likely a server-side.
I did check your homepage and do a little edit (change a period to exclamation mark), the saved take effect and I did not encounter the issues that you showed in the clip nor on the screenshot, though I see a couple of issues that both could be the culprit of what you're experiencing.
First, when you open your homepage either on Cornerstone or on the frontend, this is the very first error that you'll see on the console.
This error seems to have something to do with YouTube video and security. Try removing the YouTube videos you have on your page and observe if the issues persist.
Second, you have the PHP version 7.0 which is fine, but your MySQL version (v5.5) is behind from what WordPress require, please contact your hosting if they allowed you to update the MySQL version (depends on your plan). If not, then clone your site on a staging server with a MySQL version 5.6 or higher.
Let us know how it goes,