If you fit a multilevel mixed model, either the intercept or the slopes can vary, but the model fit at each level (and accuracy of predictions on held-out data)

If you fit a multilevel mixed model, either the intercept or the slopes can vary, but the model fit at each level (and accuracy of predictions on held-out data) will improve substantially if the terms matter and you include them. In this case I would only believe the result if it led to across the board improvement. Contrarily, if a stratified fit's cross-county predictions were dramatically shittier than within county, I might believe it, but by pooling information in a Bayesian framework these authors appear to have aimed at a happy medium. The multilevel aspect is important (country -> county -> locale) because of the shrinkage you get as a side effect of propagating each estimate. If time permits I will look at their code and data (available in the supplement, the author appears to have used Rstan for the fit, although a Python person could probably get a similar result with PyMC or PyStan) and try fiddling some of the dials. The write up is a little wordy, but the model and data is all there for anyone genuinely interested in sensitivity analysis or alterntive fits. They're not hiding anything that I can see.