I had a separate tablet with me on the trip so I quickly searched to see if this is a known issue and if there was any easy fix. Searching on Google, I found out that it's what people call the Green Screen of Death so it's not unheard of, but nobody can fully explain the cause. The general consensus is that this is a hardware failure and requires the phone screen to be replaced. The speculation is that it is caused by the phone overheating, but whether it's a defect in the design or manufacturing is still under debate. I don't have any definitive answers but will share my experience here. My situation started after I left the phone in the armrest of my rental car for a couple of hours one morning. When I took it out and turned on the screen, it started flickering and then turned dark. After a few seconds, the screen came on only to flicker and turn dark again. The phone felt warm but not burning. It might have felt a bit warmer than when it is doing a fast charge.
On YouTube, there are a number of videos that offer a "fix" to the problem, but none of them mention that their fixes are just temporary. Their "fix" mainly is to press down on the screen at different locations starting with simple taps and if that doesn't work then move to more firm presses with your thumbs until the screen comes on. Since that's really the only solution I found (besides the "Call Google and get a replacement"), I gave it a try and it kinda worked. Sometimes the display will come on but inevitably it will go dark. I'm not sure if the pressing thing really helped or if it was just that the display just turned on while I was pressing. I guess the theory is that the screen lost some kind of contact inside the body and pressing the screen will help it regain that contact. Are there that many moving parts inside of a smart phone for something to come loose due to heat? I think it's more plausible that the heat damaged some components so I don't think the phone will recover on its own.
Until I get the phone fixed or replaced, I wanted to make sure that the phone can continue to function enough so I can transfer the data over. That meant keeping the phone somewhere cool and minimizing moving or using it. The screen seems to work longer when the phone remains cool. I tried to charge it one time and when it warmed up while charging the screen started glitching more quickly.
Not having a mobile phone these days is really inconvenient. We rely on it for maps, photos, and so much more beyond phone calling, but I guess a positive outlook is that it forced me to really disconnect from the grid while on vacation. This experience also made me aware that Google has pledged to make their phones DIY repairable so you can now buy genuine Google parts to replace the screen, battery and camera. That gives me a bit more confidence in getting another Pixel. My experience with the Pixel has been good and I usually noticed the battery capacity dropping after a few years rather than any hardware failure so being able to replace the battery (even though it doesn't look super easy) is better.
I also learned that you can get the phone fixed quickly at a ubreakitifixit shop if one is nearby. What I worry about sending a phone to a shop is that unless the phone is completely wiped clean then it might be possible for the shop to access the data on the phone. These days we have so much data on the phone and if the problem is a broken screen then you can't even wipe the phone before sending it in.