With regards to the body, there is no question: the D7500 is far better suited to fast action photography. The D500 would be even better, but is perhaps outside your price envelope.

The choice of lens, as you've surmised through good research, is less clear. All of your concerns are valid.

Personally, I would lean toward the 80-200mm. The 70-300mm is too damn slow for action photography in low light. Its VR is of no help for that use case. And although its AF-P focus motor is indeed very quick, it isn't clear to me how well this particular lens's AF works when the subject is moving quickly and the camera is constantly adjusting focus to compensate. Might there be a performance reason none of Nikon's pro zooms use stepper motors for focus?

The 80-200mm is not only much faster in aperture, it's a pro lens through and through. The two-ring version you're considering is pretty fast in the focus department. Granted, that depends greatly on how powerful the focus motor in the camera is—and this would be my only area of concern, since the D7500 is a "prosumer" rather than a pro camera. I just have no data on how fast its focus motor is. But screw-drive AF speed is not only dependent on the camera; it also depends on how quickly the lens is geared and how precisely its geartrain fits together. In that area, I would trust that the 80-200mm D is good enough for sports. I think the same is true regarding its sharpness.