Basically, the result of an experiment has to be boiled down to a single numerical value, called the test-statistic. Typically the test-statistic is a (log) likelihood ratio. It is the distribution of the t.s. that must be considered when determining the significance of a measurement. Obviously the measurement itself only gives you a single value of the t.s., so you need to know the distribution to ask "does this result seem significant?". This is done by considering all the factors of random variation (statistical and systematic) that could have an effect on the t.s. Often, the distributions of these individual random factors are assumed to be Normal, but the resulting distribution considering all of their conspiring effects is very seldom normal distribution. Even in the central limit theorem, I think the distribution of the LLR ends up being something like a noncentral chi^2 distribution.

Especially since, isn't this an average / error of a mean estimate? So even if individual observations are non-normal, this would be a perfect place for Central Limit Theorem.

I know nothing about Quantum though, only maths.

Because it's a non-linear world? And the graphs seem very obviously skewed? And it's kurtosis also seems to differ from three?

None of those things matter to the central limit theorem.

If I have IID observations with finite 2nd moment (variance), then their average will pretty quickly converge to a Gaussian distribution. And I can relax a lot of this and still recover a variant of CLT.

Of course maybe the calculation is different, eg it’s not like there are N independent observations, but rather some other complex condition solved for the mean estimate.

Also not knowing anything about this topic, I'd assume it wasn't normal because we're talking about mass close to zero, and mass must be greater than zero.

The mass in their result is 80433.5±9.4 MeV/c^2. The result of the experiment is a Gaussian like distribution. If you consider a Gaussian distribution with μ=80433.5 and σ=9.4, the probability to get a result that is less than 0 is 4E-15899105.

I filled this widget https://www.wolframalpha.com/widgets/view.jsp?id=53fa34c5c66...

And got this result https://www.wolframalpha.com/input?i=mean%3D%5B%2F%2Fnumber%...

Note in the graphic that σ is 10000 smaller than μ so the probability to get a negative result is almost zero and you can just ignore it.