This is also right I believe, normal distributions are not ubiquitous really, just they are approximately ubiquitous (and only really if "ignoring rare outliers", and if you also close your eyes to all the things we don't actually understand at all).

The point on convergence rates re: the central limit theorem is also a major point otherwise clever people tend to miss, and which comes up in a lot of modeling contexts. Many things which make sense "in the limit" likely make no sense in real world practical contexts, because the divergence from the infinite limit in real-world sizes is often huge.

EDIT: Also from a modeling standpoint, say e.g. Bayesian, I often care about finding out something like the "range" of possible results for (1) a near-uniform prior, (2), a couple skewed distributions, with the tail in either direction (e.g. some beta distributions), and (3) a symmetric heavy-tailed distribution (e.g. Cauchy). If you have these, anything assuming normality is usually going to be "within" the range of these assumptions, and so is generally not anything I would care about.

Basically, in practical contexts, you care about tails, so assuming they don't meaningfully exist is a non-starter. Looking at non-robust stats of any kind today, without also checking some robust models or stats, just strikes me as crazy.