You actually need quite a bit of money to be eligible for the diversity visa AKA green card lottery. One of the requirements is that you are able to support yourself for weeks/months while you are looking for a job. Proof of funds is required, and it better be around $10k per person.

Having said that, I still think you are on point about the American Dream available for everyone willing to work hard. It just starts before you enter the US since green cards are not handed out that easily even though it is called a lottery.