Not every billionaire uses their money for that kind of power, but some do, and it hurts democracy when they do. In fact, it's even been argued that legitimate philanthropy is undemocratic, because it's rich people deciding which problems society will fix, rather than a democratic process.
If money only provided luxury, people would have less problems with this inequality. And people would probably be less interested in amassing more wealth than they can consume.
What a lot of people don’t seem to understand is that propaganda doesn’t cause beliefs, it focuses and amplified beliefs that preexist at least in nascent form. The Koch brothers would have no power if there weren’t people ready for their kind of message. In fact this is a good example. The Koch Brothers were against the Trumpian turn in Republican politics and spent heavily against it. How did that work out for them?
Sorry money == power is too facile.
The fact that some rich people only get part of what they want because they're opposed by other rich people, does not mean that their money didn't get them power. The Koch brothers got a lot of what they wanted.
And even if poor people, if there are enough of them and they are sufficiently organised, are able to oppose rich people, that still doesn't mean that the rich people don't have more power individually. Their resources make it much easier to organise, to buy influence and support, to reward politicians who legislate the way they want.
In the US, even more than in other countries, money is absolutely power.
If there were no billionaires would the so called "power imbalance" be restored? Does someone with $900 million not have the power of someone with $1 billion? Of course not.
And money doesn't linearly equate with power. There are a number of powerful people in Washington, even unelected ones, who have no substantial fortune to their name yet wield far more power than say Jack Dorsey, Bill Ackman, or Marc Andreeson. If Piketty was truly against all concentrations of power, he would call himself a libertarian.
But Piketty is not against concentrations of power. The core idea behind Piketty's philosophy is a moral one. He believes that some people having a significant amount more than others is a moral failure in society, and must be corrected, even at an economic cost.
The problem isn't merely the concentration of power, it's the concentration of unaccountable power. This is also why people are upset about secret courts and lack of congressional oversight over secret services: it withdraws accountability from an institute wielding power.
I specified the qualifier "unelected"
What Piketty and others of this philosophical movement believe is that taking money away from people is the same as removing power.
They are going to be disappointed. Power is a capital item, not a revenue item. Money gets you the contacts with the right people. Lack of money doesn't remove them.
Plus, of course, there isn't a fixed amount of wealth - particularly not financial wealth. So why confiscate when you can accommodate?