mobile.weather.gov then save the bookmark to your homescreen. Boom. Information straight from the source (National Weather Service) and it 'appears' like a regular app. Honestly surprised how good their mobile page looks.
I have no idea why you are saying there is no such app. The Google Play Store shows both the NWS logo and the NOAA logo on this app. One can also go to mobile.weather.gov in a mobile browser and directly access the NWS.
The Weather Channel is part of a for profit corp that sensationalizes weather in order to increase ratings. I don't believe for a second the stats that someone posted earlier that show the weather channel as having the most accurate weather. They get most of their info from the NWS anyway.