> The essential statistical question that remains is to ask whether this correlation is simply a staggering coincidence.

> In other words, I have no idea of how to provide a statistical likelihood that this is a realistic behavioral model of the driving physical mechanism behind ENSO.

The usual way to do this is to split the data into training and testing sets
(Really one should split into training, validation a test, but train & test will do here).
Use the train set to fit your model and and then use the fitted model to make predictions on the test set.
If the predictions are still good then there is grounds to say the correlations are real.

The simplest way to do this is to pick a date such that there is enough data on either side of it.
Fit using the data before the date and evaluate on the data after that.