I don't think you can make that assumption, many people are able to put their personal beliefs and views to one side and act professionally.
Fair as far as legitimate political views are concerned, not applicable if and when such views extend
outside democracy, making them illegitimate for policy or action; for example, holding that religious law supersedes civil law in the public sphere (or private, in the case of how children/the defenseless might be treated). In this case, such a one should not hold any public office or discharge any public duty, as they are
in danger of following their conscience which, in this case, would be a totalitarian act. Similar for good ole Southron boys if they regard rights as extending to one or select group(s) alone.
As actual examples of policies that, in practice, violate the basics, we have the Stand Your Ground law in FL, which in the Trayvon Martin case, was basically used to lynch a completely innocent black man and let a white man go, one who later showed exactly who he is, a violent racist. As implemented in the justice system, it is blatant institutionalized racism. Other policies would include no Sunday voting and national elections held on work days: racism dressed as "Good ole boy done thunk somethin' up, and we gotta do it, cuz gawd sed so; ain't racism, who you callin' racist?"
Someone who drastically disagrees wrt tax policy or how to address social issues is no problem; no conflict with foundational concepts, no fundamental paradox. But I most certainly disagree that trusting to a general capacity for extraordinary objectivity - loyalty to duty above beliefs
that supersede that duty - is practicable in reality. Good example there is the drift toward Christian Taliban fundamentalism now blessed and encouraged by SCOTUS, such as in the Hobby Lobby case and its plague of suddenly limp-wristed cakebakers and county clerks who'd rather die than comply with civil law.
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To stay on OP: The expression "Bully Pulpit" used to be used often. It's now being stricken from the lexicon by -- what else -- eternally panicking whine boys. Can't have anything around that might induce clarity, by golly.