First of all, the problem of the preview should be fixed. I should insist that the last resort will be you to migrate the WordPress to your local machine or another server to test the case, We established that you do have a dedicated server, but the problem might be the settings or Firewall which I can not confirm at the moment.
I checked your website and disabled all plugin except the Cornerstone. The result is indeed much better. I managed to add a test page and used the Cornerstone without any errors:
The same page gives the preview error when the plugins are on. You need to do the tests by enabling plugins one by one and check the test page to see which one is causing the conflict. You need to change the test page and save to make sure everything is working ok.
Now regarding the Homepage, the homepage still throws the error while all plugins are disabled. I installed the WP Server info plugin which you can check from Dashboard > WP Server Infor. The Max execution time is slow on your installation. Kindly contact your hosting service provider and ask them to change it from 30 to 300:
The Memory limit is ok and 512MB seems to be sufficient, the Homepage does have many sections which might be the cause of the problem. I suggest that you install the Duplicate Post plugin and clone your homepage and try to remove the sections one by one. See when the preview comes back.
Needless to say that even when you have preview problem you can use the Skeleton Mode to change things around in this case:
Another point is that the cache plugins might have a transient cache which in some cases cause the problems. I suggest that you install the plugin below and clear the transient cache on your database:
I also added the code below to your functions.php file of the child theme to avoid the W3 Total cache throw false error while checking the Cornerstone:
add_filter( 'w3tc_can_print_comment', '__return_false' );
Please kindly consider that we also here try to find why this is happening in your case, that is not happening on our servers and our local installation. Most of the time the Memory limit is the problem, in other cases the execution time, in some cases a firewall of some kind. Unfortunately for your case, we could not find the reason for the problem n your homepage.
The last point is that the plugins are all disabled now as I do not remember which one was enabled before. You need to enable them one by one and test the test page.
Thank you for your understanding.