Can fixed-effects become biased due to random structure misspecification I'm following-up on this great answer. Essentially, I was wondering how could misspecification of random-effects bias the estimates of fixed-effects?
So, can the same set of fixed-effect coefficients become biased if we create models that only differ in their random-effect specification?
Also as a conceptual matter, can we say in mixed-effect models, the fixed-effect coef is some kind of (weighted) average of the individual regression counterparts fit to each individual cluster and that is why fixed-effect coefs in mixed models can prevent something like this Simpson's Paradox case from happening?
A possible R demonstration is appreciated.

Heterogeneity alone does not constitute a random variable.  It is the act of sampling that does.  If, for example, you have collected data at various sites and decide to treat site as a random effect you are saying that these sites were randomly selected from a broader population of sites and individual patients represent repeated measurements on each site.  Site is now the primary sampling unit.  The endpoint and regression coefficients could be interpreted to pertain to sites instead of people.  If you did not actually randomly select these sites from a broader population, that is you would not imagine using an entirely different set of sites each time the experiment is performed, then site should not be treated as a random effect.  If sites are not considered random, any inference we derive is on a broader population of subjects who visit the fixed collection of sites in the study.  We could include site as a fixed effect but this makes the results site specific which may also not be of interest.  If we treat sites as random anyways this can most certainly introduce bias in estimation and inference.  It all comes down to the sampling scheme.
