The reason for this limited power plan set, and annoying habit of appearing offline after a period of inactivity, is the implementation and application of Modern Standby (S0 Low Power Idle). As you might expect, this reduces the power consumption of your device when it detects limited use, but does so at the expense of certain peripheral devices, including network adapters.
Open an administrative command prompt and enter the following command to see if Modern Standby is enabled:
powercfg /a
If the returned information contains the following, then the traditional power states (S1, S2, and S3) are turned off and only S0 applies:
The following sleep states are available on this system:
Standby (S0 Low Power Idle) Network Connected
Hibernate
Fast Startup
If you want to turn off Modern Standby, and have the other power plans available, you can do so by adding a new DWORD to the registry. From the admin command prompt, run this command:
reg add HKLM\System\CurrentControlSet\Control\Power /v PlatformAoAcOverride /t REG_DWORD /d 0 /f
Restart the PC then check the Power Options have been restored (and the machine should now abide by your instruction to remain accessible).
To revert back to Modern Standby, simply delete the registry entry added previously:
reg delete "HKLM\System\CurrentControlSet\Control\Power" /v PlatformAoAcOverride /f