There are two issues, IMO, regarding the press/media.
1) the trauma Harry experienced due to his mother's death and the role of the media/paparazzi in that trauma. It is not only the accident itself, but also the aftermath in the boys being trotted out in front of Balmoral to view floral tributes to their mother; the walking in the funeral cortege with their dad, uncle and grandfather; and the general press intrusion/interest in the lives of the British Royal Family.
Harry has come through all of that with a very cynical view of the role the press plays in the popularity of the BRF as well as how the BRF interacts/negotiates with the media - and it is understandable to an extent. The media can either splash those negative stories all across the broadsheets or they can hush them up. He has, IMO, experienced this from both sides of the equation - the Nazi costume, the drugs, the Vegas stripping incident are the negative, but the cheeky, Harry the Lad, Invictus Games etc are the positive.
Ultimately, he has come to believe that the BRF have far more power and control over the press and what is printed about them than is true. I suspect he very firmly believed, by the time he and Meghan started dating, that all it would take was some strongly worded letters from Buckingham Palace, Clarence House and Kensington Palace to those media entities to put an end to the truly terrible (and racist) headlines. The truth is, that the relationship between the press and the BRF is symbiotic and while I'm sure Harry grasps this, considering what he wrote in Spare about Camilla, he absolutely refuses to accept that this is the way it is and he alone cannot change it.
That leads us to
2) both Harry and Meghan want to control their press coverage. They do not WANT the negative stories and, I'm quite sure, came to believe, that they knew better than the KP, BP and CH communications teams. They chafed at the negative stories and Harry convinced himself that there were other, nefarious actors within the BRF at work, aiming negative press attention onto he and Meghan in exchange for positive press attention for those others (Charles/Camilla and William/Kate).
So, their grand plan was hatched to fly the coop and set off for America to control everything, except they didn't expect to be cut off financially and have to really do a lot to earn money.
Now, Meghan understands how the Hollywood press machine operates, and she's perfectly fine in that environment, except that they're, right now, leaning into the victimhood culture and how aggrieved they are. Which would be fine if they really were done that wrong, but every time they've opened their mouths in the last 2+ years to tell us their "truth", they've run up against fact-checkers who are more than willing to call them out on any exaggerations or outright lies, especially when it comes to the BRF as an institution.
And, now, we have multiple incidences of their "truth", well... varying from other people's recollections. Both of them still seem very determined to keep on sharing their "truth" and attempting to control the press. I'm not sure when, if ever, they are going to realize that this is futile. There will be positive stories and there will be negative stories. The best you can do, as a public figure, is cultivate good relationships with those journalists and you do that through your staff - both communications and security and learning how to tune out the ugly stuff and just get on with the work you find important, empowering and relevant.