I think this is a big fuss over nothing. If they’d followed the usual rules of titles and styles, if Harry and Meghan had needed to use a surname after their marriage - which they didn’t, because Harry is a Prince - the convention would for the Duke and Duchess of Sussex to use Harry Sussex and Meghan Sussex - i.e. their highest title would become their surname.
Archie and Lilibet would follow this convention, at which stage male primacy has an impact. As oldest son, Archie would take his father’s second title, Earl Dumbarton, as both a courtesy title and surname (so he’d be Lord Archie Dumbarton) Lilibet doesn’t get a courtesy title, but as the daughter of a Duke she’s entitled to style herself as Lady using the familial surname, even if she’s the only one in the family who uses it! So she would be Lady Lilibet Mountbatten-Windsor, like Lady Sarah Armstrong-Jones was the daughter of Lord Snowden (Anthony Armstrong-Jones), and the sister of David Linley (David Armstrong-Jones, with courtesy title Viscount Linley)….
BUT the entitlement to Princely titles changes all that. Prince Archie and Princess Lilibet would not traditionally need a surname. Modern life has moved on though, and they do really need one for official documents and school enrolments etc. It has became the norm for Princes/Princesses to adopt the designation of their parents’ highest title as a surname (so, Harry Wales, George Cambridge etc - though he might be George Wales now, we don’t know).
I can’t even get exercised about the fact that Meghan still uses Markle, so has no need of Sussex. The youngest Mitford sister, Deborah, married Lord Andrew Cavendish in the 1940s, and due to the death of his older brother in WWII went through several different titles and styles and ended up as the Duchess of Devonshire. She mostly signed her name in correspondence as “Debo Devonshire” but wrote books as “Deborah Mitford Duchess of Devonshire” - so it’s all a bit of a free for all!