Setting automatic timer for Xbox games

Please access Set screen time limits for your kids on Xbox | Xbox Support to set an automatic timer for you console.

Also see if there are parental settings on the console. For instance, there are gaming timers so that only so many hours can be played on the Xbox under parental controls, and also "break reminders" for if you're on it too long. I would check in the settings
to see if there are any automatic timers set to turn the console off.

Have you thought to check your family settings on the console? It's quite possible that there is a timer set up to automatically shut down/disconnect.

Start a pyramid game, open menu and options while in game, timer on/off should be the first in the list. When you change it, a new game with the new setting will automatically start. Timer should appear on top right.

One game should be enough for the achievement (I got it years ago so I cannot verify this last point directly).

Look in the parental settings see if a timer is set to shut the Xbox down, or your Xbox power supply is connected to a timer plug, more likely the parental setting