Corporations literally buy laws via lobbyists. Capitalism doesn't have to encourage corruption: it's already corrupted.
Maybe you could think of some other system where personal connections isn't the only source of power, but so far democratic capitalism is the only thing we have.
Edit: As for your scenario with capital paying money to influence politicians. The alternative is that the politician already have all the power (and money), nobody has to pay them. To me it is better that there is some corruption with money going to politicians rather than politicians just having all the money from the start.
Both elected and unelected officials are so corrupted that they take money from the corporations to choke the competition on the free market. To fix this issue, let's give them even more power. That will show them and they'll stop being corrupted.
Lobbying is not democratic. Corporations have far more money than individuals and therefore far bigger lobbying power. As a result, laws and policies will reflect what the small elite wants rather than what the people want or need.
Because of lobbying, the copyright monopolists have systematically robbed of our public domain rights. Because of lobbying, we might as well not have the right to reverse engineer anything.
Lobbying is literally billion dollar industries asking the government to help protect their interests. Hard to think of anything less democratic than that.
When you can't earn more by working fairly then people start to work unfairly, meaning they start taking bribes or perform other corrupt acts. Current capitalism has problems but at least the system doesn't encourage corruption. Some people will still be corrupt, of course, and some companies will work like a communist state and limit income for its workers encouraging those workers to become corrupt, but corruption will always be way worse when you put national caps on income like this.