Apologies for the Daily Mail reference, but it's all I have so far.
Apparently PC told H&M shortly before the Oprah interview that his intention was to amend the Letters Patent so that only the children in direct line to the throne (aka William's children) would receive the Prince/Princess title.
So neither Archie nor Lilibet were going to be "upgraded" to Prince/Princess on the Queen's death.
That explains that slightly vague comments made in the OW interview about Archie not being made a Prince; and why relations between PC and H&M were clearly very fraught.
If this is true (and I accept it's hard to see the wood for the trees with the constant press briefings from both sides) I can see it from both points of view.
Harry is in Prince Andrew's position (as a royal, not as a person) which makes Archie and Lilibet on a par with Beatrice and Eugenie. As they get older, and William's children start their own families, the Sussex children drop down the pecking order pretty quickly.
I think Charles has it right that the British Public don't want to keep paying for an extended royal family, you can see that playing out in Europe at the moment. So even if H&M were working royals, their children likely won't be; and I can understand why he wants to present a limited number of royals to the public.
From Harry's perspective, it must feel like a downgrade. It must be difficult being the second (and only other) child in a hierarchical monarchy; being aware from the beginning of all that William is to inherit. Having his children lose something that is currently their birthright must sting a bit.
This should probably have been considered before William and Harry had kids. I doubt it would have been as big a deal when you are not considering actual children, just abstract ones. Or at the very least, it should have been done when the Letters Patent were altered for George, Charlotte and Louis.
It's clearly added to an already tense atmosphere and won't help family relations get back on track.