While I am a biologist by trade, not a statistician, I still do a fair bit of statistical modeling. It seems like you were taught a flawed, or at least incomplete understanding of mixed effects modeling. While your experimental design is what determines whether an effect is fixed or random (at least it should be), the description you gave doesn't really match with how they are used (at least in my experience with modeling). I'll try to break that statement down a bit just based on what you wrote in the question.
... my understanding was that a fixed-effect model is when you are sampling an entire population, for example, a school district, a school, or a village. And a random-effects model would be used when taking from any of the populations a sample where the goal of the random-effects model would be to estimate the population.
First, you don't need to sample the entire population to measure a fixed effect. If you surveyed 10% of the people in a village at random, you still model your response variable as a fixed effect for that village. But, say instead you sampled 10% of the household units in a village, sampling each person within each household once. There are likely to be various environmental or genetic factors that are shared within a household that vary between different households. Because of that non-independence, each set of household samples can be thought of as it's own sub-population of samples that aren't truly independent from one another. So you would include a random effect term for households in your model, to account for those variations. But the actual response variable (or variables) of interest would still be modeled as fixed effects.
The same goes for the example in the paper. Even if the entire population had been sampled, each individual has their own unique set of genes and metabolic processes that can vary from person to person, so those repeated measures are not independent from one another. Modeling individuals as random effects allows you to see how your variables of interest are related at a population level without being confounded by those little difference between people.
This is one reason that cross-sectional study designs, where each subject is sampled only once, can be limited when it comes to generating generalizable inferences about a larger population. Without those longitudinal samples, it's not possible to account for any of that inter-individual variation in your models.
Mixed models can get pretty complex pretty easily with more comprehensive sampling efforts and study designs. Say you have longitudinal samples collected from multiple households across multiple villages. Now you have to consider how all of those confounding variables (village, household, individual) are related to one another and design statistical models in which each of those terms are appropriately nested to properly account for random effects.
For some more reading, I think this chapter from Applied Statistics for Experimental Biology by Jeffrey Walker does a pretty good job of explaining mixed how models work with some good examples of how they are constructed for different scenarios.
