I think this is coming from Harry and he's probably going to be a bit surprised that he has far less control of the press outside of Britain. While the British tabloids print lies all the time, in the wake of Diana's death, the British press were legally restrained from pursuing the royals, especially their kids, as they went about their daily lives. They still are. There are laws restricting how close they can get and what sorts of long distance photos they can take.
Those laws are nonexistent in the US and Canada and so, it is unlikely he can do much about the pack of photographers outside their rented home or those that will pursue them in the car or on foot. It is only if they create a hazard like causing a collision while following them that they can be stopped. I think he is incensed that photos of his wife and child walking in a public park were sold and published, but he has no grounds for a lawsuit. If the photographers work for a British publication, he can try to get them on the grounds of violating the British laws concerning photographing royals in private moments, but I'm not sure anyone has tested that outside of Britain.
And, every time he and Meghan and the baby leave their home, there are going to be civilians with cell phones snapping away too, and some of those will be published.
To me, it has become increasingly clear that Harry has been really triggered by all the press coverage, especially since Archie's birth and he's reliving all his own childhood trauma as well as the death of his mother because of it. I hope he's getting counseling because it seems to me that, if anything, his actions over the past few weeks have intensified the scrutiny and it isn't going to let up anytime soon.
It seems that Harry wants to be a public figure only when it benefits himself and his causes but not when it doesn't. It doesn't work that way, and no matter how many threats he makes, it never will.