I went to a grammar school, my kids go to a private school. Not because I think it's better academically but for the sport (in my kids' case).
I don't think you can easily emulate the extra sport. My kids play club and county sports as well as school sport, so it's on top of, not a substitute for out of school sport. It's the facilities, the number of fixtures, the extra S&C training.
But I think you can get a lot of the same benefits independently. I've done a fair amount of work with my kids, so for example, doing some more advanced maths papers or creative writing. My friends at state schools pay for tutors, although I think you can do much of this yourself (certainly for primary and some of the less technical GCSEs).
Agree with the visiting different places - museums, the Old Bailey (free), walking around cities, places abroad etc. For older kids, getting them to seek interesting work experience opportunities.
But overall, I think a lot is down to aspiration and expectation, which isn't exclusive to private schools. I'd say setting high standards and working hard are the key differentiators whether you're in a state or private school.