I also keep getting variations on the following question: "I bought the Z# with the kit lens. Should I sell the kit lens and get the f/2.8 zoom, or buy the f/1.8 primes?"
I don't want to seem condescending, but this is a "not sure what I'm doing" type question. Those questions are never answered the way the questioner expects them to be, because they aren't representative of a real or known problem. It's more FOMO (fear of missing out).
We have a lot of new Z camera users now, as this was a relatively successful Christmas for Nikon's mirrorless system, and we had three new models to attract people who formerly sat on the bench. So let me turn these questions around and suggest an approach to finding the answer yourself.
- Just shoot with the kit lens for awhile. There's not a dud in the lot. The 16-50mm, 24-50mm, 24-70mm, and 24-200mm are all excellent lenses as far as they go. Sharpness is not typically going to be an issue with them, and chromatic aberration shouldn't be a problem, either. I'll happily use any of the Nikon Z kit lenses for general purpose work and not worry about it. We would have been extremely happy with any of these lenses a few years ago, now we're just happy with them ;~).
- Assess issues that arise, and correct. If sharpness is an issue with these lenses, then you need to figure out why. It's one of two things: your handling and understanding of the camera, or you got a bad copy of the lens or camera, or both. A lot of people think that VR means every picture will be sharp, for example. It doesn't work that way. First, VR doesn't halt subject movement, and second, if you get really sloppy with your handling thinking stabilization "will fix it", VR very well may not be able to compensate. You also may not have mastered the focus system yet. As I suggest in my books, start with simpler, static, one-time focus situations and work your way up to complex, dynamic, continuous focus situations. In doing so, you'll likely verify for yourself that the lens probably isn't an issue.
- When you're sure the lens and camera are performing correctly, look for issues again. This is where you start to discover that the f/6.3 maximum aperture on the lens is keeping you from isolating subject from background, or that star fields aren't rendering as points in the corners of the frame, or the linear distortion correction is not rendering corners quite as well as the inner field. In other words, in this step you're trying to identify a specific trait that is causing you grief. Once you've identified that, you can start to evaluate what other options might help you correct it.
Yes, the 24-70mm f/2.8 S is a better behaved lens than the 24-70mm f/4 S. Most people won't be able to see that, and certainly not without pixel peeping. Ten years ago, the kit lenses weren't so great. These days, the kit lenses are quite good, and you have to pay big dollars to get small gains over them.
Finally, lenses have purpose. The purpose of the kit lenses is to be the Swiss Army Knife in your collection: general purpose. If you're filleting fish or carving a turkey, you probably want a different knife. So what is the photographic thing you're trying to do that requires a different knife (lens)? For instance, portraits generally need a 70-105mm focal length and a reasonably fast aperture to create the more studio-type shots you're imagining. The 85mm f/1.8 S is a great lens for portraits. But it's not general purpose ;~).
I'll likely have more to say about this as I try to rationalize my own Z lens gear closet in the coming year. Some of my F-mount lenses on FTZ adapters are being retired, some aren't. But it's always about purpose.