Have you written a serious game engine? Parent is right, a bunch of them basically are a real-time OS sitting on a small amount of the underlying OS (
if one exists). When you have to write your own malloc() and fopen(), and your own thread scheduling, that counts as Operating System. This is becoming less true today, as the hardware gets better and bigger and faster and resources are less constrained, but it wasn’t that long ago that many console games literally were the OS.
Your analogy and definition is one reasonable viewpoint, that the OS is the thing that bootstraps software on top of hardware and provides access to resources like memory, filesystem, network, and display, but it’s also an incomplete definition. An OS provides a standardized environment for whatever processes run inside it. In that sense it doesn’t matter whether or not it’s contained in a different OS. Ubuntu doesn’t suddenly become not an OS when you run it in a VM, it still provides the Linux OS to the processes that run inside the VM. Windows doesn’t become not an OS if it boots on a hypervisor. Similarly, even if an OS requires another OS to run, when it provides a different environment or container, it’s still serving the purpose of providing a specific Operating System to the software it runs.
Some consoles used to not provide an OS at all, the game provided all of it. The developer was responsible for handling things that would (and should) otherwise be the purview of the OS or drivers, power loss, out of memory, cables yanked, etc., otherwise they would not be allowed to publish. Nintendo did this longer than the others, in my recollection of events.