Is there any technical or substantive reason to typically want to ensure the location $\mu$ and precision $\phi$ parameters in the beta regression model include the same fixed or varying predictors?
I wouldn't be aware of any such reason.
I think it's a reasonable default to assume that all predictor variables that affect the mean $\mu$ of the response variable actually affect its entire distribution and hence also the precision $\phi$.
However, it is not unusual to have predictor variables that have a clear influence on $\mu$ but less or virtually no influence on $\phi$. Also, it is conceivable that you have predictor variables that are explicitly associated with the precision $\phi$ (as they correspond to "uncertainty" or "ambiguity" in some way) but not (much) with $\mu$.