I think it's more structural rather than mental.
To me, the NBA is equivalent to the jump from college (or hs) to the real world. If you have parents/siblings/family who have already operated in the NBA (or the real world), the expectations, the structure is different than if you're the first in your fam to jump into that world.
There are other examples, but the best comparison is that of my experiences relative to my wife's. I was the first in my fam to go to college and therefore the first to do that transition from college to a white collar world. My wife's family is full of doctors, attorneys, professors, & generals. Wife's first job was a friend of her fam's. I found mine off of Craigslist after a number of interviews off of CareerBuilder. She went to Cal. I went to UCLA. So the job finding process, all things being equal, should be relatively the same. To lessen the commute for her first job, her dad bought her a condo. I found a sublet, for a dirty ass room, living with a bunch of Sac State dudes. Then later, when she was considering a PharmD, her dad offered to pay for it. While when I was considering b-school, I had to wait a number of years before I could afford to go.
Someone like Klay, after college, his dad likely provided structure that would take any other rookie, from a poor environment and no NBA dad, years to figure out. Some do, some don't, some fail then figure it out later.
Now, to be clear, having structure and privilege doesn't guarantee success. There are plenty of kids from privileged backgrounds, who end up being in really bad shape. I know for a fact that one of the biggest worries of parents in the Penninsula (Palo Alto, Woodside, Atherton...) is the suicide rate. Kids in private schools or ultra competitive public schools where the peer and parental expectations are through the roof.
No magic bullet, but, yes, once kids of former players have gotten to that NBA level, I think they tend to overshoot expectations. Not just because they're mentally more ready, but because they have structural support/encouragment/expectations--guidance on where to prioritize their time that others without former NBA dads do not.