I think that the problem of the public not understanding economics is very similar to the problem of people not understanding personal finance.
There are two schools of thought in regards to solving this problem of financial illiteracy: teaching people more about finances when they are younger, and increasing the supply of financial advisers. Both of these solutions would solve the issue of people not handling their finances well, but both of them also have consequences.
If we teach people more about finances in school, theoretically, they can handle their finances better throughout the rest of their life. However, the time that they spent doing that could be better spent learning about something that truly interests them, something that they can offer the world.
If we increase the supply of financial advisers (through whatever method seems most appropriate) that people can hire, then people can increase their overall wealth by gaining from the trade of their money for the advisers service. However, people will have to give up their independence to handle their finances if they need to hire an adviser to do it.
This dichotomy holds true for the economic literacy and democracy as well. If we spend our time and resources on educating the public more in depth about economics our opportunity cost is a public that could specialize in other departments. If we give more control of government to highly educated economists we decrease the democratic voice of the individual.
So with these solutions we have to ask ourselves: How much democracy are we willing to give up?