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This has been in my head for a while, and I think I finally have the answer. I was wondering what was so special about the Carnot cycle so it was used to demonstrate the Carnot theorem; the demonstrations I had read didn't used any property specific to the Carnot cycle, but used a reversible engine. Why couldn't it use the Rankine cycle, for example?

And now I've realized that the Carnot theorem hasn't anything to do with the Carnot cycle, does it? It just states that the efficiency of ALL reversible cycles will be the same, but the demonstration talks about an hypothetical reversible cycle, not necessarily the Carnot. Am I right?

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Yes, you are right. The Carnot theorem says that all reversible cycles must have the same efficiency and any irreversible cycle must be less efficient; whereas the Carnot cycle is just a simple example (arguably the simplest possible) of a reversible cycle. The Carnot cycle is often used to motivate the Carnot theorem but there's no fundamental connection between them other than that.

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