The same propaganda machine that labelled Obama "a socialist Muslim Kenyan agent and anti-Christ" also assassinated the character of Bill and Hillary for 30 years (including attributing something like 40 murders to them) and later credited them with the "under age sex trade ring" (queue PizzaGate and the QAnon canon). Then they attacked Obama's birth certificate. The same thing happened with Obama's advisers (eg. Valerie Jarrett).
It's a power play by politically-biased news sources which convince people of conspiracies. The details are irrelevant because the conspiracies that don't gain traction are left behind and the propaganda evolves. The only details that matter are the ones that best apply to Confirmation Bias.
The NYTimes has a podcast called RabbitHole[1] which describes how this works so well on the internet and reverse engineers how a YouTube visitor was radicalized and later rejected his previous radicalization views.
The fact that there are a plethora of conspiracy theories in the ideology and that it's largely people who identify as Christian or are pro-Christian seem tangential to me.
Were the Obama religion conspiracy theories being driven by "Christianity is the norm", or by Islamophobia? Suppose, counterfactually, that instead of a nominally Muslim father and stepfather, his father and stepfather had been Jews or Hindus or Buddhists? And, likewise, suppose, that instead of spending some of his childhood in majority Muslim Indonesia, he'd spent it in Israel or Nepal or Thailand? And, suppose instead of an Arabic-derived middle name, he had a middle name derived from Hebrew or Sanskrit or Pali?
An "Obama is a secret Hindu" or "Obama is a secret Buddhist" conspiracy theory would have had far less steam, because whatever misconceptions the average American likely has about those religions, far less Americans fear them than fear Islam.
And "Obama is a secret Jew" would have struggled because it sounds antisemitic, and antisemitism is far more taboo in American culture than Islamophobia is.
Anti-Muslim sentiment in American culture is largely specific to Islam, most of it is not driven by some generic "hate everyone who isn't Christian" sentiment.