Alexander, “propagation of error isn’t related to standard deviation v. standard error though, it is how you translate uncertainty through a function of what you’re measuring.”
In my opinion you need to consider both what you are measuring and what that measure is intended to represent. If you are looking for a metric to represent energy, you have the zeroth law to consider.
That leads to the “unforced” variability or how much the temperature metric can vary with no change in the energy it is supposed to represent.
So while propagation of error isn’t related to sd v se for a particular data set you need to consider how that data set can impact overall propagation of error. That is what leads to SD being preferred.
All of this focus on “surface” data is due to the “surface” data not meeting expectations. Since the expectations are energy related, Wm-2 and Joules, T has to be thermodynamical relevant, meaning you have to consider that pesky zeroth law.
For Karl et al. time will determine how useful it is, then we should be into ERSSTv6beta. Or everyone could consider that for a planetary scale problem you need all the data you can lay your hands on and the combination will give you a better estimate of the real uncertainty. Picking a choosing probably isn’t the way to go.
