I solved most of the mystery above. It was not really a CloudLinux performance problem, it was a ISP networking issue and a contributing factor was WordPress did not make it easy to see what the problem was.
One of the WordPress plugins was trying to check for updates and the ISP was blocking the IP address of the plugin site. WordPress hung and then timed out. A second try clicking on “installed plugins” produced the same result. Each timeout took about 3 minutes each, then after two timeouts the “installed plugins” plugin page worked fine.
The IP address should not have been blocked, however, WordPress didn’t give any error or warning either, which made it very hard to troubleshoot. I’m running WordPress 4.5.2. This is a good candidate for an enhancement for WordPress.