Harry is of royal birth, which brings with it privileges as well as prohibitions, but so too were his cousins, Zara & Peter, Beatrice &Eugenie, Louise and James. At one time they were in the line of succession and received some protection which I believe ended when they became adult and were not working royals, this being one of Andrew's many gripes. The cousins all seem to have found comfortable private lives with occasional publicity. This was what Harry appeared to want when he chose to leave the inner circle of his family.
His desire to be 'half-in, half-out,' of the working royal family was refused by the late Queen, and endorsed by the King. Harry's refusal to accept his loss of status and accompanying privileges forms the basis of his grudge against his father. His vulnerability is compounded by his foolish behaviour in publishing a book detailing his activities in Afghanistan, against Armed Forces protocol, while ensuring he receives maximum publicity for his (and Meghan's) activities here and abroad. Unless of course, they suddenly demand privacy; the recent will-they won't they visit to London just highlighted their rudeness.
His obsession with the Press stems from his mother's death, avoidable had she followed the bodyguard's advice.
Harry's mental health issues stem more from his limited understanding of the fact that royal birth does not carry immunity from normal standards of behaviour (as Andrew discovered), that actions have consequences, and an inability to accept that he is the author of his own misfortunes.