Chromecast with Google TV doesn't have TrueHD support, ethernet, 6/8ch PCM HDMI output, automatic frame-rate switch, game streaming (yes this is an obvious troll, but still it is true). nVidia Shield have those, and more.
Also, the Nvidia Shield is truly a lot better in terms of hardware and application support. Quite a lot of Chromecast "apps" are inferior to the Android TV versions that also run on the Shield.
I think you're misinterpreting the meaning of the location permission: it's probably requesting that to connect to the Chromecast via WiFi Direct or Bluetooth. Android buckets that into location because you can figure out a user's location by looking at nearby WiFi networks/Bluetooth beacons.
[0] https://9to5google.com/2020/11/25/google-tv-netflix-dropped/
[1] https://9to5google.com/2021/03/11/netflix-directory-comes-ba...
The Netflix app has been working the entire time on the "Chromecast with Google TV" device, which is what cprecioso recommended.