If I inherit a billion dollars tomorrow, I will have zero additional information and be no more sophisticated than I am today. But I will have deeper pockets than any retail investor and will be able to withstand market irrationality longer than them.
But we aren’t talking probability. Your claim was that deeper pockets means you are more sophisticated. That simply isn’t true. The inheritance argument is just one example to show why it isn’t. People make large amounts of money all the time in one field or another, but that doesn’t make them sophisticated investors.