Contrary to OP's belief that "those sources [are] not generating normally distributed electrical observables", in that source first studied with respect to 1/f noise -- the vacuum tube cathode's trapping sites -- an individual trapping site releases electrons according to a simple exponential relaxation law N(t) = N₀exp(-λt). If we extend the normality concept outside the Gaussian distribution conformance to just simple ubiquity of the observed law of power spectrum, the Lorentzian power spectrum N₀²n/(λ²+ω²) of relaxation law can certainly be considered a "normally distributed electrical observable". Finally, in an ensemble of trapping sites with the relaxation rate parameter λ uniformly distributed over the ensemble, the resulting noise power distribution is 1/f.

The comprehensive review of the research into the phenomena see in 1/f noise: a pedagogical review by Edoardo Milotti

A spoiler:

12. Conclusions

... do we have by now an "explanation" of the apparent universality of flicker noises? Do we understand 1/f noise? My impression is that there is no real mystery behind 1/f noise, that there is no real universality and that in most cases the observed 1/f noises have been explained by beautiful and mostly ad hoc models. ...

One thing that disturbs the imagination of those struggling to understand the 1/f noise phenomenon, is the divergence of the total energy of this noise. The spectrum diverges logarithmically both at the lower and upper frequency limits. There is no evident problem of this kind with white noise, because the white noise is always bandlimited by well known physical laws: for example, the thermal noise obeys to Planck's law of blackbody radiation and, at room temperature, exponentially decreases to zero at frequencies above few terahertz.

The cited reference calms down the divergence worries with very ingenious argument. Let there be some unknown physics under this noise's origin that permits the exact conformance to 1/f

¹⁶, so f ~ 10^-17Hz. The smallest observable time, Plank's time, is 10^-43s, so F ~ 10⁴³, and the integral totals to 60/0.434. The measurements in thin-film resistors shows near perfect 1/f noise over more than 6 frequency decades and provide an experimentally measured confirmation for one-tenth of our theoretically estimated total. The divergence, due to logarithm behavior, is not that frightening, as it seems at first glance.