I am a bit late to this thread. Whether you will like the1.4x TCIII may depend on several factors. On the gear side, it depends on the lens you attach it to and the camera body you are using. It may also depend on how close you are to your subject.
The 1.4x TCIII will generally work better on lenses that are better optically and focus faster. I like the 1.4x TCIII a lot on the 500 mm PF, 300 mm PF and 70-200 mm f2.8 E FL lenses. Less so on my 200-500 mm lens (which I sold last a year). The 500 mm PF is a bit better optically and faster to focus than the 200-500 mm. I thought the 1.4x TCIII made the 200-500 mm slower to focus, although this may have been more related to the DSLR bodies.
I also find that I like the 1.4x TCIII better when used with a Z mirrorless body (currently the Z6II, Z7II and a recently added Z9) than with my DSLRs (currently the D500 and D850). There are a couple of reasons. Using the 1.4x TCIII on a lens that is f5.6, like the 200-500 mm or 500 mm PF, makes the combination f8. More recent NIkon DSLRs (including my D500 and D850 and I believe your D7500) will AF with a lens that is f8. But there are compromises. First, you will find that only a small subset of AF points near the center of the frame will work. On a mirrorless body, all the AF points continue to work at f8 (and even f11) and cover the entire frame. I also think that TCs may slow AF more on a DSLR than on a mirrorless body, although I have not done quantitative testing on that.
Another issue that I find better on mirrorless bodies and TCs is that you generally do not need to focus tune a lens +TC combination on a mirrorless body, given that AF is on the sensor. This can be an issue with TCs on lenses on DSLRs even if the lens does not need tuning when used without a TC.
All in all, I like the 1.4x TCIII a lot with my 500 mm PF, but prefer to use the combination on a mirrorless body.
Good luck with your choice.