This is a shot in the dark, but it's possible your HDMI cables aren't fully supporting the 4K. I was having trouble with a 4K monitor not displaying properly and it turned out my HDMI cable was outdated. Replacing it with a DisplayPort fixed the issue. That being said, most modern HDMI cables should support 4K, but it's always good to check the hardware specs of the cable and the TV.

That was my first suspicion. The cables definitely support 4K. I enabled diagnostics overlay on my FireTV and the resolution was definitely 4K and yet the picture looks washed off.

Probably a software issue - the TV is reading the incoming SDR signal and treating it as Rec. 709 (the SDR standard) as opposed to Rec. 2020 (the HDR standard). Since the color primaries for Rec. 2020 are much wider, the values in the signal representing the "amount" of each color are smaller. If you then treat those numbers as in the SDR gamut, you get a washed out (less saturated) image.

This is also just a shot in the dark, but matches your observations pretty well.

Whoops, this should read "the TV is reading the incoming HDR signal". Just wanted to clarify.

I can almost guarantee that the issue is your Fire TV's input is set to a specific picture mode, whereas the built-in apps are using the color-inaccurate vivid mode.

Go to settings, all settings, program tuning and on the first option there press 11111 and a diagnostic screen will show HDMI settings.

It might be because of different display settings applied to different inputs.