I updated to Android 12 yesterday, and appear to have runn into the same problem as everyone has reported above. It was late night, so I did not feel like troubleshooting. Connected to USB C charger as a workaround.
Looking around this morning, Settings/Display/Screen Saver/Use Screen Saver is toggled to Off. The "When to Start" option is set to "While docked".
I searched in Settings for "stand" and did not see a result.
Searching found
https://support.google.com/pixelphone/answer/9137565?hl=en and the "Change interruptions & brightness (Pixel 5, 4 & 3)" section may have a clue about going into Settings/Connected Devices. In Connected Devices my Pixel Stand shows in the "Previously connected devices" section. Not sure why that is. The Pixel Stand "Settings\Screen off when dark" is toggled off, and "Do Not Disturb while docked" is toggled to On.
I did notice last night that the Pixel 5 definitely detects the stand upon placement in the stand. The charging screen/symbol briefly shows, and then the screen stays on, and then shows the Lock Screen wall paper and clock (large font size). When I tried this it was past 2300hrs, so that was past the schedule time range I have set for "Bedtime mode", which is 2230-0600 Sunday through Thursday.
I've listed more than one variable (Setting/option) above, so may experiment with one variable at a time over the next few days to see if anything changes. Another option could be to remove/forget the "Previously connected" Pixel Stand, add it again, and see if anything changes, or if I am presented with an option(s) to setup the phone/stand so that the screen goes dark as with Android 11.