What you are talking about is basically the Lamarckian picture of adaptation. It was considered dismissed, because there didn't seem to be a plausible mechanism for encoding and passing on such changes, except reproductive success. Epigenetics, and phosyphslation of genes impacting their relative activity, has changed that.
There is also Multi-Level Selection, the modern development of Group Selection, widespread scepticism towards which I take it explains your very cautious and tentative phrasing. EO Wilson is a hundred times the scientist Dawkins is, and makes a very well evidenced case that mechanisms which can act on social insects can also be relevant for humans. See discussion here: Are humans becoming more hive-like? Does this have philosophical implications?
"Now they [genes] swarm ... safe inside gigantic lumbering robots"
-in 'The Selfish Gene'
And the phenotype-robots are controlled by our neurotransmitters regulated by genes, whereby if the incentives change we might cease to reproduce before we can evolve to meet the change - like The Giant Jewel Beetle That Mates With Beer Bottles, which was seriously damaged by the similarity of an Australian beer bottle to it's evolved mating cues (they changed the packaging to prevent risking it's extinction).
Eminent biologist Denis Noble has a more optimistic counterpoint to Dawkins:
"Now they [genes] are trapped in huge colonies, locked inside highly
intelligent beings, moulded by the outside world, communicating with
it by complex processes, through which, blindly, as if by magic,
function emerges. They are in you and me; we are the system that
allows their code to be read; and their preservation is totally
dependent on the joy we experience in reproducing ourselves."
-in 'The Music of Life'
If we look at the emergence of intelligence, we find play among all the brightest creatures, the exploration and development of capacities, and a restlessness towards repetition and stasis, that makes boredom painful especially for young organisms. You can make a good case that having by far the longest period of development spent incapable of self-sufficiency, and with development neuroplasticity until typically age 25, that play and adaptation to the social environment through it, are the defining behaviours that set humans apart. See points of evidence and discussion here for more: Video games as new art
Language is a good example. Babbling and the 'language drive' is clearly mediated by the FOXP2 gene complex, with any damage to it resulting in severe language defecits. The gene complex is also found in song birds, and mammals with complex vocal communucation like elephants, bats and beluga whales. Yet, humans raised without other language users around them until after they reach teen years, like children raised by wolves, seem to have become incapable of fully developing human language use (these cases are admittedly very rare, though). See an account of children intentionally raised without speech, where it seems they babbled instead in visual signs we might call proto sign language: According to the major theories of concepts, where do meanings come from? This seems a good example, of how genes matter, but the environment the genes encounter matters too, and a set of traits around play and extended neuroplasticity matters too. We know young children learn languages relatively effortlessly, before their brains have fully developed.
We know there are behaviour complexes that had major impacts. The emergence of new stone tools other than handaxes marks the shift from the Old to the New Stone Age, and perhaps relates to overcoming the crisis of the Toba Catastrophe eruption. The Bronze age technology suite involved 'beaker ware' clay pots, more sophisticated fire use for furnaces, and brewing beer; and this set of ideas seems to have spread far more quickly than the people who developed them dispersed. Cities and writing emerged exactly and only and in at least three independent locations, where animals of burden had been domesticated. It hardly need saying that marked one of the most significant in what humans do. These all seem to be examples of memes and meme-complexes, with interactions with genes like by domestication, including self-domestication by humans (shorter guts, less aggressive, etc.). Through all of recorded human history there have been regional contentions for power, and those have tended to come down to which human group has the most sustainable access to resources, which generally had meant biggest trading network, pushing for increasing collaboration and cooperation as more crucial to success than competition.
Nicholas Christakis talks about the evolution of a 'social suite' which seems to be the basis of our morality, in this podcast on
Humanity, Biology, and What Makes Us Good. Or Jonathan Haidt et al argue for evolved Moral Foundations. These are examples of sets of evolutionarily relevant impulses, that seem to help enable social organisation. In particular the regulation and manipulation of shame and disgust, seem to be core to what we call culture, as discussed here: How do ethicists tackle the question "Is it immoral to have sex in public places?" Is it possible to use rational and empirical ideas to answer?
I would point to the formulation of core ideas and 'psycho technologies', as crucial to understanding the modern era. Rights discourses, jury trials, the end of almost all absolute rule and of feudalism, freedom of association and of the press, are things which have huge impacts of countless lives once adopted.
We know humans have developed the capacity to avoid jewel-beetle like behaviour through cultural change, though admittedly we don't always see it, perhaps because social change has accelerated. See the example of addictive drugs, for how lack of wellbeing is a far better predictor of harm than drug availability, plus how societies adapt to harmful behaviours, discussed here: What if heroin was served instead of alcohol?
There is a very real risk of getting the entirely wrong impression that we can reasonably and rationally govern every impulse as a fully independent individual. Many thinkers have pushed back on this over the last half century, like scepticism towards 'homo economicus' free rational individuals in models, to Wittgenstein on language-as-use, yo Donald Hoffman pointing out we cannot trust evolution to show us reality as it is.