For some reason, I watched the interview last night. It seems that the point of contention was that their children wouldn't be titled as Royals so they wouldn't be eligible for security protection. I don't know how the system works. It might be that Meghan doesn't have a royal lineage. I seems that there are people who "handle" the Royal Family and their relationship with the public that were telling Meghan to stay out of the public eye when she hadn't left their house in months. She said she was to the point of thinking everybody would be better off if she was dead. She contacted one of the handlers to see about getting some mental health help and was told that wouldn't be possible.
They don't seem to have a problem with the Queen, but a problem with those handlers. Harry's children could only be in line for the throne if something happened to his brother William and his Williams children but I suspect there is some duty to maintain a strict Royal lineage. Someone discussed with Harry if he had considered how dark skinned his children might be. The tabloids of course are constantly vying for readers by trying to dig up dirt. Meghan was an easy target because she was divorced, of mixed race, has no royal lineage and is older than Harry.
There may be another component at play behind the scenes. It has long been rumored that Harry is not the son of Charles but was fathered by Diana's riding instructor. Someone would know for certain from DNA but that would be an embarrassment to admit at this point. If they can use Meghan to disqualify their children as heirs to the throne, it never has to become an issue.
Harry said he had talked to his father Charles about what was going on until Charles stopped taking his calls and told Harry to put it in writing.