RE: I tried installing the code snippets plugin and it crashed the site each time. I had to delete it through sftp for the site to come back.
Ok, the point of this test was to eliminate Connections as the cause of the issue you are experiencing. The Code snippets, like many plugins including Connections, create custom tables and need that core WP file in order to create them.
For some reason this is not possible on your site. Do you have any security related plugin installed on your site? If you do, perhaps it is blocking access.
It might be worthwhile to open a support ticket with your web host too. I suppose they might have strict security policy in place where that file is only permitted to be accessed under specific conditions. Just guessing.
By chance, is your web host GoDaddy? I found this thread in their support forum.
I also found this in the wp.org support forums.
Maybe one of those links will help.
I hesitate to recommend this, especially if you have having permissions issue. You could try reinstalling WP. Do this by clicking the
Re-install Now button on the WordPress Updates admin page. That could possibly install all the core WP files again with the correct permissions.