I didn't realize that James was elevated to a peerage on his own. My mistake
You cant compare James to Charles, William or Harry. He doesn't hold a peerage. They are The Duke of Cambridge, The Prince of Wales, The Duke of Sussex. And even then their full title is HRH Prince Charles, The prince of Wales..... Prince William, Duke of Cambridge. Prince Harry, The Duke of Sussex. Simply they are referred to by their peerage.
Yes Charles is referred to as simply The Prince of Wales. But his full title is HRH Prince Charles, The Prince of Wales...….
William is HRH Prince William, The Duke of Cambridge
Harry is HRH Prince Harry, The Duke of Sussex
Archie is not if he had a courtesy title, some nameless Earl. He would be Archie, Earl of Dumbarton. It's not like his name would not be used until he became a prince. Just like William is still called Prince William and not always Duke of Cambridge.
How they are commonly addressed even in the CC is not the same as what their actual name/title is.