your TP buttons should definately be working, regardless if the ATK driver and/or TP drivers are installed or not, as windows (especially win 10) will install its own driver for this, so this maybe be a hardware issue.
And the images you provided in regards to volume and TP disable are from the older style ATK driver package (Win 7-8.1).
By rights, you should be able to download and install a newer version of the ATK package, even if its from a newer ASUS ROG gaming laptop that was released with WIN 10,
example
I am using the ATK driver (Version 1.0.0039) from G750JM
found hereon my G74sx (which is older than yours) with Win 10 pro. all is working fine.
Head to "Programs and Features" (WinKey + x) then F key will open it,
Uninstall the Current ATK package first, then reboot.
Once rebooted, install new downloaded package and reboot again.
You should be able to do the same with ASUS Smart Gesture TP drivers too if the newer laptops use that TP, but from what im seeing, on asus, your laptop uses Synaptics TP, so you should try Synaptics drivers.
To check which TP is used,
1.Open Device Manager (WinKey + X) then M key
2. expand "Mice and other Pointing Devices"
right click on ps/2 port conpatible TP (if your using an external mouse you might have 2 options) and choose properties/details
in the drop down menu, you will see "Device instance path" and "hardware ID's" and "Compatible ID's".
if you see "SYN" in any of those, you are using a Synaptics TP.
Alternatively you can copy the value from "Device Instance Path" and search the NET to see what if comes back for that ID.
Unfortunately ASUS has dropped Synaptics TP from their G series line so no more updates to the drivers, latest from ASUS for G75vw is v16.XXX.
so the best/latest Synaptic driver i could get working on my sys is version 17.0.18.0, they have v19+ now, but Synaptics have been really slack with their drivers, not signing them properly etc.
IF you do have the Synaptics TP, and cant find the 17.0.18 driver on the NET (not on ASUS, or Synaptics Websites unfortunately) PM me and i will upload to Google Drive for you.
and it was hard to find, just keep digging.
Hope this helps.
See how you go.