Can you replace the battery with an extended one?
That's a deal-breaker for me, and battery cases are not a good replacement since I can't rely on the phone's own battery indicator, have to manually turn on/off and looking at ZeroLemon's case for the S7, it seems to require a headphone extender.
As for the S3's screen I have no complaints. It's readable in the sun with max brightness, I keep it at minimum the rest of the time.
Finally, the S7's native screen resolution is 1440p, meaning it will munch on battery much faster for no perceptible gain (at the distance I hold my phone). And manually lowering the resolution will render a blurrier image than native 720p so that's out.