I think I understand how supercomputers work in general, but I would like to know how they are configured. It seems to me that supercomputers have a separate computers at each node, each with it's own processor, memory, storage, networking and operating system. It's all that really necessary? Why isn't each node given a processor and memory and a way to communicate with a master node, which is a real computer. Each with be provided with binary ready to run. Why does each node need to be able to run it to own operating system.
It is pretty hard to come up with a specific definition of a supercomputer. From what you described I think you have clusters in mind, which is basically a network of computers with some task-division possibilities. Every node in a cluster is a computer itself and they might (or should) communicate with each other.
I think your question indicates a slight lack of understanding what computers are and what operating systems do (what do you mean by real computer?) If you have a processor and a memory you need some operating system to use them effectively. For example, you mentioned "a way to communicate" between machines - that is pretty tough to implement reasonably without an operating system.
I would recommend checking wikipedia page on computer clusters and taking it from there.