Here's the SOLUTION so far:
Tech support at LiquidWeb (our webhost) found that by disabling the PHP extension Suhosin, Grade Publishing worked.
I do not know if disabling Suhosin could cause other problems.
The server that was giving the problem with Grade Publishing had these specs:
CentOS 7, 4GB RAM, PHP 5.6.25, MySQL 5.6.33. It contained all the PHP extensions recommended in the Moodle installation.