The 24-120mm f4 is a very good (but not great) lens indeed but consider the facts that it costs much more and on your D5200 it will be effectively 36-180mm lens so you'll miss out a lot of the wide-angle range. The 17-50 f2.8 should suffice for most of the occasions not to mention its a full stop faster and also optically considered a bit better on DX. The range from 50-120mm (or 75-180mm equivalent) is usually in no man's land unless you want to shoot portrait in which case you need to get 85mm prime. If you are sure that you don't need the longer tele range, then you can get a wide zoom like Tonika 12-28mm or Sigma/Nikon 10-24mm along with the 24-120 but that will increase your budget by a lot.
Once again, the 24-120 is a pretty good lens for FF camera but it doesn't make much sense for a walk-around zoom on a DX. If you are really not willing to change lenses, I suggest to sell your current gear and get a Sony RX10 (
currently @ 55k) - it has a great 1" type sensor and a fantastic 24-200mm f2.8 Zeiss designed lens that should cover you all your necessities unless you shoot wildlife/birds.