When do we need an Operating System in Embedded System Design?
My rule of thumb is that you should consider an operating system if the product requires one or more of the following: a TCP/IP stack (or other complex networking stack), a complex GUI (perhaps one with GUI objects such as windows and events), or a file system.

If you've done some bare metal coding then you're probably familiar with the super-loop program architecture.  If the product's firmware requirements are simple enough to be implemented with a super-loop that is maintainable (and hopefully somewhat extensible) then you probably don't need an operating system.

As the software requirements increase, the super-loop gets more complex.  When the software requirements are so many that the super-loop becomes too complex or cannot fulfill the real-time requirements of the system then it is time to consider another architecture.

A RTOS architecture allows you to divide the software requirements into tasks.  If done properly, this simplifies the implementation of each task.  And with task prioritization an RTOS can make it easier to fulfill real-time requirements.  An RTOS is not a panacea, however.  An RTOS increases the overall system complexity and opens you up to new types of  bugs (such as deadlocks).  As an alternative to the RTOS you might consider and event-based state machine architecture (such as QP).

If your product has networking, a complex GUI, and a file-system then you might be at the point where you should consider full featured operating systems such as VxWorks, Windows, or Linux.  Full featured operating systems will include drivers for the low-level details and allow you to focus on your application.