[noggin]

It's not framerate matching but my TV reports BT.2020 HLG HDR and it doesn't look washed at all.

That's interesting. 

I need to see if my Sony + Denon combo (both of which are supposed to be HLG-friendly) are flagging HLG compatibility via EDID etc.  I know the combo will flip into HLG mode if I send HLG infoframes from the Vertex (so they accept HLG flags over HDMI)  However the Sony TV supports 1080p120 via HDMI but doesn't flag this via EDID - so it wouldn't surprise me if Sony were a bit behind in their EDID flagging for HLG?

EDIT - creating a custom EDID to spoof the Fire TV Stick 4K gets it to output HLG (though still Rec 709) - so it looks as if the XF9005+Denon isn't flagging HLG support via EDID potentially. Will try without the Denon in the path. 
EDIT 2 - bypassing the Denon AVR the results are the same. Looks like the Sony may not be flagging HLG support via its EDID - even though it happily recognises HLG Infoframes. If I spoof the EDID fed to the Fire TV Stick 4K to support HLG, it outputs HLG and the Sony TV switches to HLG EOTF.  But still Rec 709 even though the EDID flags 2020 support. Odd...
EDIT 3 - looks like although I can inject Rec 2020 Primary flags into Infoframes via the HD Fury Vertex GUI, this doesn't overwrite the Rec 709 flags in the underlying video signal. More work needed. Think this may be a Sony / Amazon not-quite-working-together-properly thing specific to HLG in iPlayer - not quite so relevant to Kodi yet 
EDIT 4 - swapping out the Fire TV Stick 4K for a Roku Streaming Stick I get Rec 2020 HLG output in the Roku iPlayer if the HD Fury adds HLG support flagging to the Sony/Denon combo via EDID - so there is still a gamut switching/flagging issue present on the Amazon Fire TV Stick 4K app. If I don't flag HLG support, the Roku outputs Rec 709 SDR - the same as the Amazon Fire TV without spoofed HLG support. The Amazon Fire TV Stick 4K doesn't seem to want to output Rec 2020 iPlayer content in Rec 2020 on my Sony like the Roku does in HLG mode

I know my Sony TV does HLG via it's internal iPlayer app, and I know I can manually select it in Picture Settings.

I've just upgraded my HD Fury Vertex to the latest firmware as I think I was missing a DV update. (DV HDR was not being flagged as DV or HDR on the previous firmware, I now get DV flagged in the first line - but I guess no metadata as this is Dolby specific/proprietary, unlike HDR10 metadata which is open)