It's worth noting that people who win huge amounts suddenly are typically hounded by friends and family to share the wealth.
What they don't understand about money is that if it's not invested, it runs out. People who give away large amounts because they are pressured to do so end up with nothing because they don't know how to manage large amounts of money.
In addition to the initial joy of winning, they are usually saddled with the long-term frustration of losing friends/family and, in the end, having nothing at all to show for it.
But the main point, which you made very well, is that money can't buy happiness because happiness comes from being happy with one's self. And as you point out, that's about what we do, not what we have.