UHD (i.e., 4k) and HDR (High Dynamic Range) are 2 different things. One is resolution (UHD/4k) and the other is color range/gamut (HDR). Do a google/bing search for HDR (Dolby Vision is another version of HDR) for more info...it's convoluted/confusing I know...all these acronyms.
One piece of the puzzle we've left out...are you using the Samsung "Evolution Kit"? I'm assuming not based on the info you've provided so far. Research that as well.
It appears that your TV (as is) is not capable of HDR (but is capable of UHD/4K) so my guess is now that you would need to buy the Evolution Kit in order to get the latest HDMI standard in order for your Xbox One S to connect properly/enable Game Mode. Don't know this for sure, but if you are indeed using an up to spec cable (and I agree with you that the one that came with the S should be sufficient) then that's the only thing I can think of that could be causing this to happen. A call to Samsung tech support would be my first COA before looking into the Evolution Kit if I were in your shoes.
FWIW: All PS4s (i.e., doesn't have to be a PS4 Pro) are capable of HDR gaming (via a previous firmware update). But only PS4 Pro is capable of "4k" gaming - used quotes b/c it's actually checkerboard 4k....don't get me started....more confusion for the consumer.
Bottom line: To be honest, I think you got a really nice TV that was manufactured 'on the cusp' of all this fast-moving technology/specifications and you may have gotten stuck in the middle so to speak. That being said, I'm pretty sure that's exactly why Samsung created the Evolution Kits to provide you with some future proofing. Again, I'd check with Samsung Tech Support to confirm all our hypotheses/your options.
EDIT: Re: Evolution Kit - Check post #789
in the official thread for your set as another user is asking "To be clear, you have an HU6840, with current firmware (1190), and you were able to get HDCP 2.2 working on HDMI 3 without an evolution kit?" ** OFFICIAL ** 2014 Samsung 4K UN55HU6840 Ultra HD Smart TV Thread
My guess now is that your TV's 'built-in' HDMI port's spec may be the issue and that the evolution kit may resolve this???
DISCLAIMER: I'm not an expert at all this stuff but I do know just enough to be dangerous so please don't take my word as the Gospel