You didn't say why you think you need a 24 - 70 or whether you intend to stick with DX. You do seem willing to spend close to or more than $1000.
If you are going to stick with DX the Nikon 16-80 VR may make more sense. Do you really need the increment of sharpness a smaller range high quality zoom provides? Do you really need f2.8 across the whole focal length range? In 2018 do you really want to walk around with a non VR lens? Or a lens that omits the wide angle range (DX) 16 -24 mm?

Personally I would not be interested in a 24 - 70 unless or until I had a high Mp full frame camera like D810 or D850.

Looking for advice.

Im thinking of replacing the 18-55mm kittens with the Nikon or Tokina 24-70mm f/2.8.

Both lenses are FF, but I intend to use it on my Nikon D5300 DX body.

According to my research:
1. The Nikon 24-70 has a sharpness issue at f2.8 but with an accurate autofocus. Tests made by lenstip.com showed only 1% of pictures out of focus.

2. The Tokina 24-70 is very sharp at f2.8 but suffers a higher amount of pictures out of focus, ~17%.

1st conclusion
From my current point of view this favors the Nikon over the Tokina as I probably won’t shoot that much at 2.8 due to the narrow Depth Of Field at that aperture.

2nd conclusion
Furthermore, test results show that the Nikon only has ~1% out of focus, which means less failed photos due to equipment performance.

3rd conclusion
Price is ~$850 for a used Nikon 24-70, but for a similar amount one get a brand new Tokina.

I’m leaning towards a used Nikon 24-70. Would you do the same using it on a DX body, or would you chose the Tokina?

Well, by my pen name you know which one I would go with. Better quality parts, all produced by Nikon and not produced by the lowest bidder. Nikon produces their lenses to work specifically on Nikon camera's and not to work on a variety of camera's. Nikon lenses are made to BETTER tolerances than 3rd party glass, tight tolerances are expensive to manufacture, there is just no way around it. Sure, third party glass is cheaper to start out with, but over time, not only will the Nikon lenses delivery better IQ, but they will be worth MUCH more when you sale it. Yes, you probably will make up the difference in price when you sell it. Third party glass loses a lot in resale. Those tolerances are really telling under extreme use over time. Yes, you do get what you pay for. Especially for that particular lens.