Earl of Dumbarton is H's title for Scotland. It is not Archie's, but he can use it as a curtesy. Just like Charles and others have titles for different parts of the UK. I don't recall at any point W&H, nor William's children using their fathers' secondary titles.
I have also seen it a lot here on MN that Earl outranks Prince. Don't think that's true. Archie is a sovereign prince.
I think one of their complaints was that they were not being consulted about anything, and someone decided to name Archie master, which is technically not even a title as that is what you call a mr who is under 18. Lili was called Miss.
Titles order : Duke/Marquess/Earl/Viscount/Baron.
Charles was quick to update everyone except the spares.
I don't believe that Edward never wanted the titles for his children, I think it was just announced like they did with Archie, and Andrew himself probably just made it clear he was having none of it. Anne's children are Philips 'clan' if you will, so I can see why they would not be titled.
I believe Edward was likely treated like Harry spare and bullied into submission, because he was supposed to have received the DoE title ages ago too.
MM has never addressed the Earl of Dumbarton title.