Would this effect, by any chance, be gone if the tv's "framerate" is made much higher?
No, it would not. Most TVs and monitors run a refresh rate of 30 or 60 Hz. This is already faster than your camera's capture rate. This effect is compounded by the fact that your box TV is almost certainly interlaced (redraws only half the image per cycle). If you were to somehow increase the TV's refresh rate to something like 200 Hz, it would increase the number of redraws per second, and thus increase the chance that the camera would capture an image during a redraw. If you want to minimize this effect, you should use a TV that does progressive scan (instead of interlaced), and also runs at a lower refresh rate.

Of course your best course of action would be to use a capture device that could take in the TV's analog outputs. Something like this.