In Sweden and Belgium, a Princess/Prince title does not need to include the princess's/prince's surname. For children who are not "of Sweden/Belgium" they simply use "Prince Alexander" and "Princess Anna Astrid", but the children retain their family names "
Bernadotte" and
"of Austria-Este (Habsburg-Lorraine)", respectively. (The latter is not "of Saxe-Coburg & Gotha" as in Belgium surnames are almost always inherited in male line; the Saxe-Coburg name will likely be used for the children of Prince Laurent's sons).
So, the kings of Sweden and Belgium had no need to create a separate title of "Prince Bernadotte" or "Princess of Austria-Este" (although I disagree with the belief that the king of Sweden could not do that).
Theoretically, I don't see why it could not; the children of the heirs could be "of the United Kingdom/of Great Britain" and the others could either be "of York/of Kent" or "(Mountbatten-)Windsor". However, Britain already makes the distinction by using "Duke of [dukedom]" for sons of heirs versus "Prince X of [paternal dukedom]" for other princes.
The Sussexes have never publicly alluded to wishing that their child not receive a royal title. As for unofficial stories of what they have said privately,
the story is that they have given their blessing for Archie to become a prince.
I would find it difficult to understand why palace sources unofficially communicated to the press that Archie would become a prince, if that was not the true intention.