DougJensen

Thanks for the compliment on the footage.
In my opinion, the FX6 is a far superior camera to the A1 for wildlife video (I own both cameras).  As you mention, the electronic Variable ND is one reason, but there are many others, including better ergonomics, better monitoring options, touch-screen AF, multiple Assign Buttons for instantly changing settings and modes on-the-fly, better battery system, ability to record 120 fps 4K on regular SD cards, better peaking and zebras, better compatibility with Catalyst Browse, and of course, Picture Cache.  Picture Cache is basically an endlessly looping buffer that allows you to record stuff that happened up to 30 second prior to deciding to hit the record button.
The A1 is a fine camera with excellent picture quality, but it's main advantage is size and weight.  Nothing else I can think of that would make me choose the A1 over the FX6 for wildlife video.
This video I recorded a couple of years ago explains Picture Cache better than I can here.
 

Videosoul is correct, this is very strange.  I spent about 2 hours yesterday matching all the settings on your camera that I could see in the viewfinder and trying to recreate your problem.  No matter what I do, VND functions normally.  So, I then reached out to one of the top techs at Sony overnight to see if he could figure it out , but he's stumped as well.
Here is our advice:
STEP 1:  First of all, make sure your camera is running the latest firmware, which is V4.0. If it is not, then you can download and install it here.
https://www.sony.com/electronics/support/interchangeable-lens-camcorders-ilme-series/ilme-fx6v/software/00259043
STEP 2:  RESET the camera:
Menu > Maintenance > All Reset
STEP 3:   Load my custom ALL File.
After I matched all the settings on your camera (at least the ones I can see) I saved an ALL File.  You should download my ALL file via the link below and load it into your camera.  It will be called "VND-TEST1" when you see it in the camera's menu.  After you have loaded the ALL file, check to see if the VND functions correctly now -- it does on my camera with those settings.   If you still experience the same issue then there is probably something wrong with your camera.
ALL FILE:     https://vortexmedia.cimediacloud.com/r/PIFxEvRjExYn
Disclaimer:  Just to be clear, none of the settings saved in the "VND-TEST1" file are settings I would ever use myself except for 4K XAVC-I.  So, if anyone else is reading this thread, I do NOT recommend the settings in this file. I'm just trying to troubleshoot Nailer's problem.
STEP 4:  Report your findings! 🙂   Good luck.

Well, that's the kind of thing that sounds good on paper, but usually turns out not so good in practice.  There's no way of creating a better REC709 scene file than S-CINETONE for the FX6 because there are virtually no paint menus to make it with.  Not that I think S-CINETONE is very good.  The lack of paint menus is probably the second biggest shortcoming of the camera. 
As for baking-in an MLUT, I've never seen it done successfully.  I'm not saying it is impossible, but I would have to see some actual real-world video that shows a custom MLUT in practice.  Until then, I'm a skeptic.  🙂

Hi Jiri, I agree completely with your comments.  I have made this suggestion to the Sony product managers several times, but it is unclear whether we will ever get our wish or not. But you never know.

I just coated my 200-600 and 70-200 in black over the holidays.  I sure am liking having most of that ugly beige covered up.
BTW, the A1 is a great camera, but for wildlife video, it is definitely a big step down from the FX6.  Every time I take my A1 out so I am lighter and more nimble, I end up regretting some missed opportunities that the FX6 absolutely would have gotten.  No ND, no picture cache, non-movable EVF, are major shortcomings.  
Just my 2 cents on the subject.

When I'm shooting with my F55 or FS7 I always shoot S-LOG (or RAW) and then grade in Resolve. Almost no exceptions to that workflow with those two cameras.
However, after doing a lot of controlled testing and experimentation with my FX6, I feel that I generally prefer the results I get when I shoot with S-Cinetone and then do a final grade in Resolve.  Now, someone might argue that if I'm going to run the footage through Resolve anyway, I might as well shoot with S-LOG3, but I've done my testing with FX6 footage, and I actually prefer grading S-Cinetone footage rather than S-LOG.
I know that S-Cinetone is promoted as being a WYSIWYG shooting mode that shouldn't require grading, but in my opinion, S-Cinetone can really benefit from some grading in post.  Here's a split screen demonstration I uploaded to my Youtube channel that shows the difference between Ungraded S-Cinetone and Graded S-Cineteone.  I prefer the images on the right.