I wonder the extent to which the problem is merely "Government should have moral people." (With the obvious caveat that those who proclaim their morality loudly are probably not the most moral.)

In the private sector there are plenty of cases where you're spending someone else's money on someone else and you care about doing a good job of it—and often without the requirement of personal mores as motivation. Banks and investment managers, for instance, basically do this. Some of course are awful, but some (loosely correlated with the ones who have been around for a while and built a reputation) actually do care about getting good results without excessive risk and without deception. Do the people who agree with Friedman's statement about government also conclude that private banks are bad too and people need to look out for their own money?