Why do big public schools always try to cover up their employees?

I'll try to cover a few of the basic points. The full answer is long enough to fill up a textbook. The worse the reputation of government schools become, the more parents you will have sending their children to private schools. In addition to reducing the power of the government, you'll have more parents demanding tax exemptions that are supposedly used for education. And like you said, there is a level of shame, but it moves beyond the school officials. Elected officials don't want a bad reputation for the schools in their districts because it could harm their poll numbers.