The Founders prohibited a religious test for office, having learned from the bloody wars of religion.
The Founders could not imagine that ideology would take the place of religion; that the same passions they saw attached to dogmas of petty sects would be possible without God; that a single ideological party with a greater reach than any religion would come to rule peoples and nations, including their own.
In America this ideology would create a fourth branch of government—the regulatory state—and its allies in the legislature would abandon legislative duties to write bills enabling the bureaucracy unchecked power; its allies in the executive abandon its oversight duties allowing the bureaucracy to set its own methods and goals; and its allies in the judiciary rule to expand its powers again and again.
It is not possible to ban ideology from government, as government is its purpose, and so, until we—or more likely our successors—can create new checks and balances, by ideology shall we be ruled despite its failures and interregnums.