They certainly would (and undoubtedly have.)

However, I would surmise that a better educated society, having learned how to apply rational thought, would reduce the likelihood of that occurring and/or would minimize the damage when it does (and it would.)

Their own benefit is 1) getting re-elected, and 2) being remembered through history as one of the great leaders.

Politics are a horrifically inefficient and error prone way of enriching yourself. To the extent that politicians value money, it's usually about getting elected again, possibly to pursue higher offices where they think they can have more impact.

Our corruption problem is not so much about money for politicians to spend personally, but to spend on further campaigning.

I'm willing to bet some, if not most, of the worst leaders in history were highly educated. Being educated doesn't make you a good person.

I'm willing to bet being educated makes you more aware of bad decisions.

OR, being educated simply means that your delusions and deceptions are on another level.

Possibly makes it easier for them to defend their delusions and deceptions as well.

When people are educated beyond their intelligence it makes them more convinced that their bad decisions are really good decisions.