And when a game is running, isn't the OS basically sitting around doing not much at all?
On consoles the operating system provides a lot of the features a game uses. For example the multiplayer components, audio chat, gameplay video recording, etc... So resources are reserved for the operating system to be able to handle these tasks.
On my PC I can use all of my cores when I play a game even though the OS is much more bloated than on a console.
I'm not sure what you mean by "bloated" in this context but when you're running an application on your desktop generally any operating system tasks are run as low priority backround threads and only when resources aren't needed by priority tasks. The OS also doesn't generally provide you much game-specific functionality that is required as a high priority when the game is running.