Democracy is a fragile thing. It's not a perfect system and it also is easily manipulated. Oligarchy comes with the coat of democracy.
Most of the time, the change of the head of state won't have any effect on the oligarch that actually rules the country as these rich people has so much influence that the president/pm won't want to mess with them. Some believe they are what holding the nations and in absence of them, the economy would collapse and the nation would disintegrate.
Oligarchy exist in almost all of democracy. There are certain group of people every politician wants to get favor from. Some politician do indeed tame and create new oligarch in their favor and such politician stays in power for longer. Like Putin and Modi.