So if I understand you right Zotac has DP as 'primary/native' which is the result of the command
and for physical HDMI output it goes via DP to HDMI 2.0 converter.
Yes, Kaby Lake CPUs don't have a native HDMI 2.0 controller, so manufacturers have to use a DP->HDMI 2.0 converter chip.
Would it make sense to buy DP 2 HDMI cable please?
I don't know. If it is some kind of odd HDMI handshake issue between the Zotac and your new TV I guess that cable could in theory fix that.