Does CAPE have strong predictive power?

Sure, with two caveats.

The first is that it's pretty much useless as a market timing tool.
(The only exception is that if the CAPE indicates that the market is extremely cheap, that tends to work out pretty well pretty quickly)

The second is that you can't judge it (or any predictive model for asset prices) by the random luck of a single end date. Like today, for example.
Compare the CAPE on a starting date to, for example, the average CAGR for all holds in the range 5-10 years later. This gets rid of the luck of the end date while keeping the precision of the valuation on the start date.
The correlation is pretty good then. Dates that the CAPE figure is compelling make much better than average times to buy, and vice versa.

Personally, I also use an improvement on the E10 used in the traditional Shiller CAPE calculation.
It is intended to be a long enough period that there is at least one earnings recession in the lookback period, but occasional you will see a 10 year earnings bull market and it will give an unreasonably high estimate of the "sustainable" level of earnings.
I use four different smoothing methods then look at a simple estimate of the four estimates, so it's not overly dependent on any one critical start date...sort of the backward-looking version of the forward-looking single date problem mentioned above.
This is just a tweak, not a fundamental change to the idea.

Using my own version of CAPE for smoothign the earnings, here are the 7-year forward results for various valuation levels for purchase dates since 1995.
The 7-year figures are the average forward CAGRs for holding period lengths of 4-10 years.
The middle figure is the average return, the figures to the left and right show the range of outcomes we've seen.

                                     Lowest  Pctl 10  Average  Pctl 90  Highest
Most expensive   5%  of start dates   -4.1%    -3.8%    -3.2%    -2.4%    -2.2%
Next            10%  of start dates   -3.4%    -3.1%    -2.0%    -0.5%    -0.1%
Next            10%  of start dates   -2.2%    -1.7%     0.6%     2.5%     2.8%
Next            10%  of start dates   -0.4%     0.1%     2.2%     5.0%     5.7%
Next            10%  of start dates    1.1%     1.6%     2.9%     3.7%     9.5%
Next            10%  of start dates    1.8%     2.1%     5.4%    10.1%    10.3%
Next            10%  of start dates    2.2%     2.6%     6.5%    10.9%    11.4%
Next            10%  of start dates    3.5%     4.2%     9.2%    12.5%    12.9%
Next            10%  of start dates    5.3%     8.6%    11.9%    13.4%    14.0%
Next            10%  of start dates   10.6%    11.8%    12.7%    13.8%    14.6%
Cheapest         5%  of start dates   12.7%    13.3%    14.0%    14.5%    14.7%

Note that the BEST about-7-year forward return starting from the most expensive 5% of the time is way lower than the WORST outcome starting in the cheapest 5% of the time.
All figures adjusted for dividends and inflation, assuming the investment is the S&P 500.

Jim