Hi @Brendan_Alexander, and welcome! In general, stats questions like this one, with no stan-specific component, are probably better suited for different forum, such as stats.stackexchange.com.
Very briefly, there are plenty of scenarios where some regularization of parameter estimates is desirable, especially in low-information settings, and regularization via hierarchical priors that include normally distributed random effects is often a very useful way to achieve this.
If the true generative process is well approximated by this random effect construction, then the resulting parameter estimates will be unbiased. In settings where the generative model is not well approximated by a model with a normally distributed random effect, then the random effects model is at least somewhat misspecified, and this could result in biased parameter estimates. Note however, that even in the nonhierarchical “fixed effects” case, you will need to make a choice about how strongly to regularize the coefficient estimates via the prior. In low-information settings, it is essentially–maybe literally–always the case that some degree of regularization is sensible.