I really struggle with CAPE based arguments. Typically people say things like "the historical mean is 17, that means whenever the CAPE is above 17, equities are overvalued!". But since I was born, the CAPE has been greater than 17 for all but about a single year, and the market has returned 10.4% over that same period. If one only bought equities at sub-mean CAPE, you would have needed to do all your purchasing for a single six month period after the great financial crisis in 2009.
The mean CAPE over the last 38 years has been about 25. So does that mean any time the CAPE is > 25 we shouldn't buy equities? Because 2013 is the last time that was the case, and the market has returned 11.6% since then. This doesn't pass the smell test. In short, CAPE based investing rules seem wildly conservative, and I'd much rather just DCA into the market ignoring macroeconomic trends.
I love ERN's analyses but the CAPE based approaches is one of the (very few!) weaker arguments that he makes, imho (along with his mortgage in retirement essay). I haven't seen many compelling explanations of why the CAPE should always revert to its historical mean. So much has changed, macro-economically, over the decades. Monetary policy is evolving. The biggest companies today sell software instead of material goods. Our economy is increasingly non-material, why would the market's concept of a 'fair' price stay the same over decades?
My background is in computational science, not economics, so I certainly don't have the academic or professional credentials to seriously critique ERN's methodology, but I'm leery of the Shiller CAPE.
Ultimately, everybody seems to agree on the fundamentals: a historically completely safe withdrawal rate is 3%. 4% is somewhat risky for early retirees. Choose something in between according to your risk tolerance, the prevailing market conditions, supplemental cash flows, etc. Trying to overfit models to a tiny dataset of something as heuristic-y as the CAPE seems unlikely to be incredibly useful.