I was out again today using Android Auto and this time I completed my route without the phone overheating. It was just a degree or two F cooler than last time when my phone shut down.
But this time, I changed the USB charging port I was using to one that was limited to 2 amps instead of using a fast-charging port. Along with keeping the phone display off, this was enough to keep my Pixel 6 Pro charged - in fact, it actually gained a little charge during the trip. The phone was still warm, but not nearly as hot as it was when I used the fast-charging port.
So, one suggestion for fellow riders who are having an overheating issue is to avoid using a fast-charging port. Kind of obvious, I know, but it did not occur to me before today.
Vinnie