Being an elite in anything requires the successful application of some kind of skill. That skill might be making money, writing books, being telegenic, winning votes, bureaucratic infighting, or any number of other things, but anyone who can be described as 'elite' got their position by doing something, and doing it skillfully.
In many cases, the skills that enable people to be elites do not add value to society. People who complain about 'elites' are most often complaining about people with these kinds of skills.
But if you got rid of all the existing elites without changing the incentives inherent in the system, then nothing would change. The people at the top got there because a certain set of skills were rewarded. As long as the system is set up to reward those skills, then people with those skills will always be at the top.
The more competitive the system, the more likely it is that the people who win the contest will have those skills, and no others. Developing character traits that are socially good, useful, and productive takes time and effort, and this means that you are less likely to be the best at the skills that let you rise to the top.
The quality of the people who rise to the top of a system should be seen as a symptom, not a cause. If bad people are at the top, that is a sign that the incentives of the system reward people who act badly. Getting rid of the current crop of bad people will change nothing.
In a democracy, the political elites are those who are best at winning votes in their state or district. A successful politician is one who devotes his or her entire existence to winning votes, and is skillful at this. If you have a complaint about politicians, then you are really complaining about the identity, attitudes, and preferences of the people who vote for those politicians.