Failure is not only about money though. So even those who are rich now, can still experience a failure from time to time. If we are talking about money, most rich were actually born rich already or they have a rich parent/ancestors. They still have struggles in life and yeah, the reason they overcome it and be more successful is because of determination and other positive traits you mentioned there.
Failures are moving in a downward position, so it is only normal for us to be weighed or dragged down by them. This is where we can lose our balance or focus but a good rest for a while can make our head clear and give us a good positive energy to try again and work towards our goals in life.
To be fair, people who come from a family who made a million dollars a year, is a rich kid who had rich family, but if they became a billionaire, that's still success and achievement. Because not every kid who grew up like that ends up being rich, which proves that it's a success.
Obviously not as good as someone who grew up in a family that barely had anything to eat and then became a billionaire, but to be fair those people barely become that rich, usually they end up doing fine, but look at list of billionaires in the world, and see how many of them had a family who made way above average, and how many had parents who were in poverty. I guarantee you, over 90%+ of those billionaires had parents that made more than the median, in fact probably more than double or triple of the median.
