For me a key factor is how the returns from a given start date are calculated.

If you take a return of precisely 10 years, then the result you get is at the mercy of the (more or less random) valuation level on the precise ending anniversary.
So, for all intervals with an endpoint in a period of volatile markets, you get almost no signal.

I personally correct for this by calculating forward returns with a smoothed end point.
For example, take the annualized rate of return for all possible holding periods 6 to 14 years later, and average those, and call it the 10 year forward return.
This gets rid of all endpoint squiggles, while maintaining the precise information about the valuation ratios on the start date of each hold.

The other big thing, of course, is the very long run trend of increasing valuation multiples. I don't think there are many very satisfying explanations for this, but one can't ignore the fact that it happens. There was a time when a CAPE of 17 was expensive, and in the last few decades it would count as cheap. So any analysis that does not include sub segments, or a correction factor for the long run trend, is going to have low apparent predictive power.

For example, someone bending over backwards to find some sense in it all would look not at the predictive power of a particular CAEY number (I say CAEY rather than CAPE because you can't meaningfully do arithmetic with P/E ratios), but the predictive power of a day's CAEY number relative to (say) the average trend earnings yield in the prior 40 years or the long run linear best fit line of date versus CAEY. i.e., rather than checking whether CAEY is above or below 5%, was it above or below the trend of CAEY through time? I did my study for only 1995 and later because, IIRC, it was in the discussion of whether CAPE had had any predictive power after it was popularized, and also because using a shorter interval of ~30 years gets rid of most of the problem arising from the fact that markets were very much cheaper a century ago. The amount of history you consider is the main determinant of what level you consider "normal".

Jim