Are you not in the US? Maybe it's different elsewhere, but my nexus 5x died a little while ago, and I only had the default warranty, not the extra protection, and it was insanely easy to get a replacement.
I filled out the form[1] (saying I had already tried the usual android-saving moves), got a call back in two minutes as promised by that form, the customer service person accepted I had already tried things and didn't ask me to do anything else, then I immediately got the email with the link to get a new phone shipped to me. Was probably less than 15 minutes of my time.
(you also seem to be talking about something else than the OP. If your phone died in two weeks that's definitely under the warranty, regardless of if you purchased extra protection).
[1] linked here: https://support.google.com/store/answer/6301527?hl=en
I was not asked to do any troubleshooting.
FWIW before it stopped working, the pixel was hands done the best phone I've ever used, in almost every category.
I'm very sorry you had that awful experience. I know it doesn't remedy your past experience, but please know myself and many other Googlers have been and still are putting a lot of work into the tools being used by the customer support reps around the globe to better help you.
Everyone on the team I'm a part of recognizes the perception you've eloquently provided, and very much care to scale with humans in the loop.
I remember once contacting Sun for info on software many years ago and received an email where they promised that someone would "reach out to me".
I didn't want to be reached out for - I just wanted some text answering my question.
So for now, I'm just patiently awaiting their response and hopefully they'll give me the green light to mail it in for a replacement or something.
https://www.accc.gov.au/consumers/consumer-rights-guarantees...
I'm extremely disappointed. I thought with their whole advertising of "24 hour support" they identified customer support as a weakness of theirs, and are trying to remedy it. Nope.
BTW in your case, you can probably just buy a new screen: http://www.digitaltrends.com/mobile/google-ubreakifix-pixel/