I'm in New Zealand, to be able to turn Cortana on and use it, I had to pick a different country.
The closest was Australia, but being a Kiwi, the thought of pretending to be a Wallabie was too much
, so I selected English (United Kingdom) instead.
It then pointed out this was different to my regional settings and asked me to confirm I wanted to do this. I confirmed and after a minute or so it set it self up asking me to pick something for it to do. Regretfully I asked it for a joke
At that point I went to the Registry and modified the key with the policy, then restarted the computer. Cortana was no longer appearing in the Search field, but just like you its processes were still running in the Task Manager.
So then I open the Windows Settings app, and searched Cortana, which lead me to the Cortana & Search settings, where I turned off the Device History, then clicked the Clear my device history button.
Then that last screen appeared which I clicked the close button on.
I then restarted computer.
Cortana was now not showing in the Search field, and no Cortana processes were present in the Task Manager.
I did all this on Windows 10 Home Edition, Version 1703 (OS Build 15063.483) which is what you have.
Perhaps you could intentionally choose a different Region in the Cortana & Search settings field, then as I did apply those other changes to see whether you can defeat Cortana that way instead.
Re: the Display Adapater. If you uninstall it [but do not check the box to uninstall the drivers], this simply removes references in the Registry. When you restart the computer, windows will redetect the hardware - in this case the graphics card - and set up fresh registry information for it, and use the driver currently in use now, or an older one, if it deems it necessary.