After spending about two days intensively researching this, I can see that cycles in the 'Exponential Time' sense do exist, but they are short lived. Using the data I displayed a couple of days ago, I stripped out the cycle lengths, both as calendar days and as fractional node values. The two charts displayed here, are the results using that data.
You can see that the 'cycles', I have now more modestly renamed 'Events' in keeping with what they really are.
Basically, they vary so much in duration, that to call them cycles is just wrong. They do in fact, generally decrease in length (calendar days) as you run through an Elliot wave pattern from end to end, but the real interesting fact, is that by seeking out reasonably easy to find events (like the bottoming of the Williams%R oscillator on daily charts), the relative (node) spacing of these events relative to exponential time, changes and shrinks very dramatically as you pass through a completed Elliot wave pattern. In other words, cycles they usually aren't, but as detectors of a completed pattern, yes, they can be. That's really not too shabby.
It's still true that exponential processes are throwing out a lot of extra events, both tops and bottoms, into the stream, so a Fast Fourier Transform is going to try (struggle?) to fit these things into linear cycles. That said, this approach still might still work well at very long time scales, where I suspect Cycles dominate more than Exponential Time effects...
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