“The variance between El Niño and La Niña becomes more pronounced in periods when the background temperature is warmer, and less so in cooler periods.”

doesn’t look right – the variances between e.g 1525 and 1825 look much bigger than say 1825 to the present.Yet the former embraces much of the little ice age while the latter is the time of the recovery from that cold period.
The green line looks like red noise and could well represent about half of the solar 2400 year cycle.
It would be most interesting to see a graph of the detrended data removing the green line trend and then see a suitable time period moving average of the residual variance followed by a maxium entropy power spectrum of that curve. Perhaps these already exist in the original paper.
I suggest that periodicities in this spectral analysis will be found to correlate with lunar orbital and solar orbital and activity periodicities and or the beat between them.