Maybe this will clear it up for you: That the
physicists are making a normal distribution
assumption is from shaky down to sloppy down
to silly down to
absurd. Making this assumption for
experimental data, e.g., the LHC data or
the gravitational wave data, where they are
hanging on by their fingernails anyway, is
close to treating the normal distribution as
some gift from God.

And, if some field makes a normal, or
mean 0, variance 1 normal, assumption, then
they should say so and, hopefully, justify
that assumption.

Your examples from Wolfram and Wikipedia
show that a lot of people have
guzzled that old Kool Aid on the normal
distribution. A few years ago I had a
date with a high school teacher, and she
assumed that test scores are normal.
My father had an MS in education and my
brother was a ugrad psych major, so,
when my brother's psych material got to
statistics, Dad taught him about
the normal distribution. It appears that
around 1900 and then for 80 years or so,
and still in some parts of some fields,
a normal assumption, without mention
or justification, was common and
remains in what you call the "context".
"Context" or
not, common or not, popular or not,
it's still an assumption, needs mention
and justification, and usually is not
at all well justified or justifiable.

Sure, with their normal assumption and, say,
five sigma, experimental physicists
get particular probabilities of Type I error,
so called p values. No question here.

The issue is the normal assumption.

There are places where a normal assumption
is quite solid. The most common justification
is from the central limit theorem. So,
take random variables
X(1), X(2), ... and assume that they
are independent and all have the
same distribution with
finite
standard deviation. Then the central
limit theorem says essentially that
as
positive integer n grows to infinity
the distribution of the sum

X(1) + ... + X(n)

will converge to normal (really should divide
the sum by square root of n).

So, where might we get some such

X(1) + ... + X(n)?

Sure, from Brownian motion where
there are many little bumps
and the bumps come close enough to
satisfying the hypothesis. So
can use this also in thermodynamics.

So, there are places where a normal
assumption is justified.

But just making a normal assumption
for essentially all experimental errors
as needed for the accepted five sigma
criterion is close to a weird religion
and not flattering to a modern science.

What physics is doing with their
five sigma criterion is a statistical
hypothesis test.
My examples of distribution free hypothesis
tests via resampling is a
better justified and more
conservative and robust way to do
an hypothesis test. Physicists might
consider using such.

Such distribution-free statistical methods
form a large field.

My example of the role of
Hilbert space is quite
practical.

If you want to explain to the common
man in the street why physics likes
their five sigma criterion, then,
sure, you need the normal assumption.
Then you should mention that there
physics is making a normal assumption.

There you might not confuse the common
man in the street with a claim that
in making this normal assumption physics
is close to drinking some swill of
boiled tails of rats and bats.

But with the now high interest in
computer science of big data,
machine learning, etc. I was
assuming that the HN audience could
and should hear the real stuff --
that in what physics is doing,
there's a normal (Gaussian) assumption
in there.