I think it has a lot to do with people in the social sciences wishing that the CL Theorem could simplify everything sampled to a single mathematical model they can have tools programmed to do magically when fed the data.
Macro and financial economists have traditionally been the worst offenders.
What you're saying is probably part of it, although in my experience that criticism can be leveled as much, if not more, at wet-lab-type biologists who eschew all but the most minimal stats.
With the social sciences, though, there's another phenomenon at play, which is that the phenomena are so abstract often that there's not really a good theoretical reason to assume anything in particular. And if that's the case, because the normal is the entropy-maximizing distribution, you're actually better off assuming that rather than some other distribution. You could also use nonparametric stats, but that has its own advantages and disadvantages.
Bias-variance dilemma and all that.
The truth is, it's hard to beat the normal even when it's wrong. And if you subscribe to the inferential philosophy that every model is wrong, you're better off being conservatively wrong, which implies a normal.
I'm not saying everything should be assumed to be normal. But unless things are (1) obviously super non-normal, or (2) you have some very strongly justified model that produces a non-normal distribution, you're probably best off using a normal if you're going to go parametric. And I think those two conditions are met much more often than we like to admit.
The normal distribution is kind of over-maligned, I think. I started my stats career being enamoured of rigorously nonparametric stats, and still am (esp. exact tests, bootstrapping/permutation-based inference, and empirical likelihood), but have grown to strongly appreciate normal distributions (or whatever maxent distribution is appropriate).
With data in hand, a skew/kurtosis scatter plot is a good way to gauge the higher dimensional distribution of your data. Another option is to cluster the variables of the data set using something like HDBSCAN and color the plot points based on cluster membership.
If you have to go guessing distributions without evidence, you're better off choosing a low k student's T distribution (for robustness to outliers) or a gamma distribution (if you think your data might be skewed).
Mind, you might find different distributions when solving diffusion equations (the Cox-Ingersoll-Ross process involves a Bessel distribution if I'm not mistaken). But the diffusion-jump paradigm is much better justified there than distributional (or even finite moment) assumptions in discrete land.
Put it this way: if finance is abusing distributional assumptions, there's money being left on the table.
And unsurprisingly there are quant funds and prop trading firms that use this very fact to make lots of money. Academic economist's love for the normal distribution is derided by pretty much any fellow real world trader I know.
Look, even if we disregard jumps as a possible term in equations, merely considering volatility to be stochastic and driven by a Brownian already gets you variances that may grow arbitrarily fast. This is without considering stranger nonlinearities on the Brownian term itself.
People are overly impressed by forceful arguments of the Taleb variety and log-log plots of empirical distributions and start parroting talking points about heavy/fat tails.
If I wasn't computerless and on my phone I would link to a paper that does the analog of the Anscombe quartet for log-log distribution plots "proving that data is Pareto/power law/etc." It's good vaccine for fat tail hipsterism, look it up.
The Tanaka equation is an example (admittedly: not a diffusion, not a case o plug-and-chug the Ito lemma) of a process driven by a Brownian with a discontinuous probability distribution. How the hell? From memory,
dTNK = dB if dB>0 else -dB
Now imagine a model with three equations.
X1 is a bog-standard geometric diffusion (we could have picked something that's chi-squared distributed driven by a Brownian from standard interest rate models) as in the 1970s Black-Scholes models. But instead of having an exogenous volatility, it has its Brownian term dB1 multiplied by a second equation, X2.
dX2 could be a standard mean-reversion equation, but again its dB2 is multiplied by X3.
dX3 is something like abs(dB3 - dB4* X3).
Voilà, an equation (X1) with a sudden break driven by the level of a mean-reverting equation (X2, which tells us volatility should come down in finite time even if it grows by a lot at times) that's set to blow up at a X2-dependent but stochastic level.
Don't get me wrong, Poisson-like jumps are very common (they're precisely the limiting process for sudden jumps) but people overstate (perhaps because they didn't really read the conditions for Ito isometry) how much a Brownian motion forces a system into normality or smoothness.
But hey, people get away with being hipsters about programming languages, why shouldn't they do that for stochastic calculus too, you know?