If extraneous emolument is unconstitutional, vetting it should be a prior requirement to candidacy. How can anyone serve, if they don’t qualify? Constitutionality is a requirement, not simply a lack of crime. The constitution can be violated through ignorance, apathy, or mental instability, in addition to criminal intent.
Constitutionality is purposeful action taken to avoid conflict. That is radically different than a lack of criminality. Constitutionality is more than the presumption of innocence. Popularity is not the qualifier. Constitutionality is.