I personally also don't think it is unfair if kids of rich parents inherit their money (or their support). After all, being able to take care of one's children is a major motivation of many people for trying to make money.
It's also well known that money is an important factor in mate choice for women.
Even without money, biology would be at work and people would be striving to find attractive mates to conceive "fit" children.
You could then also say those "fit" children are "privileged" compared to "less fit" children.
--
Anyway, as your own anecdotes highlight, if you are not rich, find another way, for example a rich person who gives you money.