I don't think there is any genetic basis for intelligence: if there was, then you would expect that the offspring of the geniuses we had in the past to have an higher chance at being extremely intelligent than other people. But this doesn't seem to be the case, as you usually never hear about the sons or daughters of such geniuses, as they seem to be like other "normal" people in most cases. I admit I'm not bringing any data to back my claim, but it seems to me that having a genius as a parent doesn't really change your chance at being more intelligent than average. You also have to consider that many breakthroughs in history weren't just the result of intelligence, but also being the right person at the right time in the right place played a big role.
There certainly is a genetic basis for intelligence. It's not even something that's up for debate academically, there's a mountain of evidence for it. We even have identified specific genes that play roles in intelligence. Almost every human trait is partially heritable, and it would be very surprising if intelligence were an exception (considering there's considerable variation in intelligence and it's very important for fitness.)
From the Wikipedia page that you linked:
"There has been significant controversy in the academic community about the heritability of IQ since research on the issue began in the late nineteenth century"
"explaining the similarity in IQ of closely related persons requires careful study because environmental factors may be correlated with genetic factors."
"The heritability of IQ increases with the child's age and reaches a plateau at 18–20 years old, continuing at that level well into adulthood. However, poor prenatal environment, malnutrition and disease are known to have lifelong deleterious effects"
"Although IQ differences between individuals have been shown to have a large hereditary component, it does not follow that disparities in IQ between groups have a genetic basis.[10][11][12][13] The scientific consensus is that genetics does not explain average differences in IQ test performance between racial groups"
And it continues in a similar way, so there is some genetic basis, but at the same time epigenetic factors seem to matter a lot.
The issue is the misuse of the term "heritable" to mean "genetically determined". Toe count is genetically determined; virtually all people are coded for 10 toes. But they have very low heritability: variance in toes is environmental. Conversely: dress-wearing isn't at all genetically determined; anyone can put a dress on. But dress-wearing is highly heritable: the variance is (almost) entirely due to genetic differences.
Even if there’s no genetic basis for intelligence, someone could (and, I’d say, probably will) make a marketing claim that they can select for it. Whatever they’re actually selecting for is going to start showing up in higher rates and who knows what the effects would be.
That's not how intelligence manifests. "Geniuses" are usually products of opportunity as much as anything. Indeed I am minded of a "where are they now" on british television of some University Challenge winners and despite being believed to be geniuses most of them had not done anything remarkable.
Intelligence definitely does have a genetic component - but again that isn't how genetics works. The "UN Man" Tabula Rasa doctrine is ideology, not science.
It's likely that this is true, but you can't actually prove it because you can't separate genetics from other things that happen before and during the time you're a zygote. Maybe it's having a human mother that makes you intelligent.
