Power Saving: Set my profile so that my computer's power settings can take effect

No. Perhaps you have selected Stop work after computer is idle for: 20 in your customization. This is an option common to the Power Saving configuration set. Selecting Maximum Output might be a better option.

Basically, I thought you might have set the profile of the laptop to use Power Saving options, when you moved from Work to the Default group.

There are 4 device profile groups:
Default, home, work, school

You configure these and set your computers as members of these groups.

Each of these profiles can have one of 5 setting sets,
Standard Settings,
Minimum Impact,
Maximum Output,
Power Saving,
Custom Profile.

While the Custom Profile setting is a configuration set in its own right, it is also the exception, as by first selecting one of the other profiles (saving and then re-entering the same configuration page) you can then use it to configure each of the others (Standard Settings, Minimum Impact, Maximum Output, Power Saving). In other words, by selecting Custom Profile and then making changes you actually change whatever selection set was previously selected.

The naming is very confusing, "Custom Profile" for example should really be "Customize Settings", if another set of settings is already selected. After all it's not a Profile, just a method of changing (Standard Settings, Minimum Impact, Maximum Output, Power Saving) to bespoke Settings, and these "Custom Profile" settings can be different for different profiles (work, home, school, default).

So, for example you could chose the default profile, select power saving, save this, then configure it again by selecting Custom Profile. If you then set Stop work after computer is idle for: to 0:0minutes, it would remove this from the power saving configuration of the default profile.

It's possible that the laptop is set to power down, install updates, sleep/hibernate, the power was not left on/connected, or the lid was shut?