Running without DXVK doesn’t mean it’s running in VKD3D.
TL;DR in Translation layers:
DirectX 12 -> VKD3D
DirectX 11 -> to Vulkan via DXVK if installed, to OpenGL if DXVK not installed
DirectX 10 -> to Vulkan via DXVK if installed, to OpenGL if DXVK not installed
DirectX 9 -> to Vulkan via D9VK if installed, to OpenGL if D9VK not installed
As said, VKD3D is only used when DirectX 12 is used, because it is a DirectX 12 translation layer, and nothing else.
If the game is running in DirectX 11, and there is no DXVK installed, it will run in Oldschool OpenGL Translation mode. That means the DirectX calls will be translated to OpenGL in WINE, which is CPU intensive and not efficient. The whole point of DXVK is removing some of that overhead by using Vulkan instead (because not only the translation is CPU intensive, OpenGL itself is as well). For DirectX 9 the same applies, except that the Vulkan translation layer is called D9VK (which in turn is a fork of DXVK so most of its options work there as well). In future versions D9VK will be merged back to DXVK so it’s one library for Direct X 9, 10, and 11.
The Vanishing of Ethan Carter (the Original) is a DirectX 9 and 11 title. So it is impossible to use VKD3D for it, because - again - VKD3D is for Direct X 12 only.
The Vanishing of Ethan Carter Redux (from 2015) is a DirectX 11 title. So as with the original, it is impossible to use VK3D3 for it.
If the game is running in OpenGL (see above why that is) you need to use a different command according to the Documentation: