Just to come back to points raised by @GJDS and @Jon_Garvey that I did not get a chance to respond to earlier:
I am not sure if this is relevant to your question, and you probably are well aware of this already, but just in case it is useful to the discussion, here are some comments.
There is quite a large literature modelling the population genetic effects of severe bottlenecks on genetic diversity in populations, by, amongst others, Alan Templeton, Brian Charlesworth, Nick Barton and Masatoshi Nei. This was partly motivated by a debate about whether or not founder event bottlenecks can cause speciation (note, the debate was not about whether or not severe bottlenecks can happen - it was about whether they drive evolutionary change). This led to quite a lot of empirical studies on natural populations that were known to have passed through bottlenecks (evidenced by past human observation and records) and on experimental populations. For example, here is a recent paper that experimentally shows that populations do much better after a bottleneck if the founding couple are outbred rather than inbred previous to the bottleneck: Szűcs, M., Melbourne, B. A., Tuff, T., Weiss‐Lehman, C., & Hufbauer, R. A. (2017). Genetic and demographic founder effects have long‐term fitness consequences for colonising populations. Ecology letters, 20(4), 436-444.
I think it is fair to say that models of the effects of bottlenecks on genetic diversity are well developed and well tested. Of course, there are inherent limits to how well we can test the long term effects of bottlenecks in natural populations or experiments, as we are limited in the number of generations that we can study. I guess this is the major problem that you were both pointing out.
Perhaps the best empirical study available to us on the effects of bottlenecks is the Lenski long-term evolution experiment. Though this has the disadvantage of being on an asexual organism, it has the advantage of having run for 60000 generations. This experiment started with an extreme bottleneck, as each of the 12 parallel populations came from the same bacterial colony. Lenski et al (1991) wrote: “over all the founding populations, there was essentially no genetic variation either within or between populations, excepting only the neutral marker.”
Recently a fantastic study was done by Lenski and his collaborators tracking the genetic changes that have occurred in each of the 12 populations that all originated at the same time with the same bottleneck.
The results are quite startling, in that very different dynamics have occured in each population. Here are the allele frequency trajectories for just three of the populations, from Figure 1 of the paper:
The authors found that the different dynamics were for several reasons, including: changes in mutation rates, periodic selection, and negative frequency dependent selection. The final paragraph of the paper reads:
“Together, our results demonstrate that long-term adaptation to a fixed environment can be characterized by a rich and dynamic set of population genetic processes, in stark contrast to the evolutionary desert expected near a fitness optimum. Rather than relying only on standard models of neutral mutation accumulation and mutation–selection balance in well-adapted populations, these more complex dynamical processes should also be considered and included more broadly when interpreting natural genetic variation.”
I think this perhaps supports the point you were making. It is a very very different system to human populations, but in many ways it should be a simpler system, and therefore easier to model. It underlines the difficulty of going from models to real evolution.
If we were presented with the twelve different Lenski LTEE populations that exist today and asked to reconstruct their past, I very much doubt we would be able to detect the fact that they all went through the same bottleneck 60000 generations ago.