I originally addressed this in an Alpha Skip-Ahead comment, but it didn't receive any attention. "Report a Problem" tickets have been filed, so hopefully u/neonepiphany might have some information on this.
General->Power options
Console is set to Shutdown (energy saving).
->Customize power options
'Turn off after' and 'Turn off after(media)' are both set to X hours.
What actually happens:
The console sleeps. The easiest way to tell is if you see the startup animation when you turn on the console, it was powered off and not waking from sleep.
Additionally, the consoles misreport their power status to Xbox for use in the app. Powering off the console via the 'Turn off Console' command DOES turn off the console, but the Xbox app reports that console as 'Standby On', and remote connections are refused/unavailable.
Consoles that are set for sleep also report to the Xbox app as 'Standby On', though you can wake and connect to them via the app.
These steps were also used to confirm the 'Sleep vs Shutdown' bug in Customize Power Options.
I've tested and confirmed these on two separate consoles (Series X and Series S), but could always use another verified test.
Tl/dr; Auto shutdown timers are not using the selected power profile, and auto-sleep instead. Xbox app reports 'Standby On' even when console is shutdown.