Of course money doesn't take away problems or guarantee happiness but unhapiness and difficult times are made worse when there are awful money worries - and serious money problems alone are enough to cause anxiety and depression and family strife.
There is a world of difference between having a nanny/nannies plus other domestic help in your home and having to get up at the crack of dawn to feed and dress your children and rush them off to nursery, and then rush home from work in order to avoid a financial penalty for picking them up a few minutes late. That's apart from the shopping, cooking, washing, ironing, cleaning, etc, that working mums (and hopefully dads too) have to fit in in their "free" time. Really, let's not pretend that the royal family is "just like us".