It is not genetic, but it is frequently inherited. In the pre-Enlightenment years, most people in the world tended to assume that religious affiliations and responsibilities were an assigned, not chosen, trait. So it is not surprising that we get confused about connections between the things one receives from family socially as opposed to biologically. Other common examples include ideas about social roles ("it's natural for..."), nationality, and "race".
As far as genetics go, there are theories out there that will predict religiosity being more prevalent among those with lower general intelligence. So in that sense it's partially genetic in a more indirect way.
I think you mean hypotheses; theories by definition have been empirically tested and not discarded. Intelligence is itself a deeply problematic concept from a scientific point of view, a largely subjective judgement that relies heavily on models of cultural expectation. One can be demonstrably skilled at a specific task, but choosing which tasks are important when making essentialist value statements about others is an inherently arbitrary process.
General Intelligence is not a scientifically problematic concept. It is problematic when people want to impose subjective value onto it, which most research on intelligence does not. Such as when people interpret "general intelligence" to mean everything of cognitive value about a person. The only people who draw that implication are those trying to create a strawman, so they can reject the concept of general intelligence, which they don't like for purely ideological reasons.
It is also problematic when one ignores the scientific distinction between a "skill" and "intelligence", which a more general ability to be able to acquire knowledge and apply knowledge in order execute a skill. Again, this blurring of the lines between skills and intelligence is done almost entirely by ideologues who don't like the idea that people differ in some way the impacts cognitive performance on a general level.
General Intelligence refers to the potential to learn and reach sound conclusions given a set of facts, and being better able acquire skills that depend on knowledge and reasoning. But whether that potential is actualized for any given domain will depend upon whether the person has the opportunity and motive to make use of their intellectual abilities to acquire that knowledge and reason about it.
The concept of "Multiple Intelligence" is actually what lacks any scientific grounding, precisely because it sloppily confuses the distinct concepts of ability to reason in novel situation and to acquire knowledge and skills given the opportunity with actually having specific knowledge and skills. Martin Gardner's notion that being a good dancer is a form of "intelligence" ("bodily-kinesthetic intelligence") is among the most idiotic ideas ever to come out of the field of Psychology and few who actually research human cognition take it seriously.
General intelligence refers to a set of very basic information processing abilities that tend to be relevant for learning and reasoning across many domains. But of course, moderate general intelligence combined with high level of exposure to and practice within a specific domain can produce the same level of skill within that domain as a high level of intelligence and moderate practice. IOW, "skill" level is a domain specific outcome that can be produced by various combinations of inputs, including 1) amount of domain-relevant experience, 2) level of general abilities to take advantage of whatever experiences one has, and 3) motivational/goal-related/emotional factors that impact how much effort one puts into applying one's ability to take advantage of experiences.
When the "skill" is for a physical action, then physical abilities come more into play, when it is an intellectual action, then intellectual abilities (e.g., intelligence) comes more into play.
Getting back to the intelligence-religion issue, there is scientific evidence suggesting that atheist have higher general intelligence, and that this relationship holds across countries. This is because the idea of God is rather absurd and cannot stand up to reasoned thought. That is why theism wanes wherever scientific knowledge is spread, the skills to reason are fostered via education, and people are given the liberty to use their reason to decide about God for themselves rather that coerced into it with threats, which is the main function of organized religion and their founding doctrines.
However, there is also evidence that atheists differ in both the other 2 types of factors I mentioned above. They have more experience with the kinds of critical thinking involved in education (more likely to attend college, and go to better grade-schools due to the SES-religiosity correlation), and are more likely to adopt a cognitive thinking "style" that actually makes use of their reasoning skills.
The latter one is refers to
studies such as this one showing that atheists are more likely to get problems correct that have an intuitively appealing wrong answer. For example, take the problem:
“A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the
ball cost?”
The answer of "10 cents" springs immediately to mind for most people, both atheists and theists alike. However, theists are more likely to just stick with that intuitive answer, but atheists are more likely to doubt their initial intuition and check it via analytic reasoning, which leads one to the correct answer of 5 cents. (note that this tendency to go with wrong initial intuitions and not check them via analysis is measured not just with this 1 problem, but several that share this property of having an intuitive answer that comes easily to mind, but is easily shown to be incorrect if the person applies a bit of analytic thinking).
The math itself is rather easy, and the theist can be shown to understand it if you give them a different problem without the misleading intuition. Plus, the research statistically controlled for differences in IQ, education, and SES. Thus, the difference is not due to a difference in intellectual ability, but in one's motivation/disposition to use one's intellectual skills and to be skeptical of one's own intuitions.
They did other studies that showed that this tendency to stick with the intuitive but objectively wrong answer did not predict what religion your parents tried to raise you with, but rather predicted the the person had strengthened those religious beliefs since childhood, in contrast to the people rejected the intuitive and arrive at the analytic answer who tended to have weakened in whatever religious beliefs they were raised with. They even used an experimental manipulation to temporarily put people into a mindset that was either positive or negative toward relying on "instinct/intuition". They did this by having people describe in detail a situation where their intuition/first instinct led them in the right direction and had a good outcome, or led them in the wrong direction and had a bad outcome.
Even though nothing in the experiment had mentioned "God", this simple difference in describing intuition positively or negatively caused a difference in how strongly people later said they believed in God. Of course, people will quickly return to their default way of thinking, so it was only temporary, but it shows a direct causal impact of being more postive toward intuitive thinking and believing in God.
Why would this be the case?
We are aware of our own and others' intentions and goals and how they shape the way we impact the world and the things we create. This makes the idea of super-human God whose intentions and goals create and shape the universe spring easily to mind and have strong intuitive appeal. Yet the fatal logical flaws of this explanation for the world and its blatant self-serving egocentrism become undeniably obvious with the slightest application of honest analysis and self-questioning. Thus, people more prone to stick with easy intuition, and not prone to analysis of their own intuitions are those mostly likely to maintain the theistic beliefs across their lifespan.
In sum, there is a good amount of scientific evidence that both general cognitive ability to reason and general tendencies to apply the reasoning ability that one has will make one less likely to be a theist and weaker in any religious commitments one was socialized to accept. Separately, there is evidence that a tendency to be religious is partially linked to genetic influences. As you correctly point out, this is clearly not some direct genetic coding for specific religious beliefs. Rather it is explained by the other research showing that both intellectual abilities and motivation-related thinking styles have some genetic influence, which then in turn influence religious beliefs.
