It's not necessarily the case that a genetic change must somehow improve the reproductive probability of the individual. It's more important that the long term survival of the genetic code is assured. So I might have a new gene combination that leads to a lesser chance of producing children, but if I somehow manage to do so anyway that same genetic code might enhance my childrens' abilities. Or my grandchildrens', etc. So the effect is expressed across a broader segment of the population in a more species oriented way. This would tend to restrain any genetic tendency for especially selfish types of behavior that produce morally intolerant acts and lead to the desolution of society. That said, societies tend toward a balance between altruistic and egoistic personality types in a dynamic equilibrium.
When Dawkins uses the phrase 'the selfish gene' I'm not sure he's actually anthropomorphizing genes but instead using the phrase as an analogy to help people understand how they work.
Individually, a gene that is successful will perpetuate itself, and genes that are the most successful are almost universal aspects of the each domain of life. Consider something like eyes.. almost every animal has them. This is because these genes are so enormously successful that animals cannot exist without them. But the genes for eyes couldn't care less about any of the organisms other genes, just that they continue existing.. hence 'selfish' gene.
In practice, genes don't actually have intent, but the analogy helps paint a picture of how they work. And the over-arching point is that the prime unit of selection is actually the gene, and not the organism.
If that's what Dawkins means then I shouldn't have given his ideas the benefit of the doubt. I assumed "the selfish gene" was simply a way to sell books by using a provocative title. As an instructive analogy I think it fails because it has intelligent folks like you utilizing the idea that genes "couldn't care less" about other genes, as if you were describing the attitude one person has towards another. You can say that was not your intent, but I see it all the time in discussions and I think it only increases the ability of people to rationalize selfish behavior. Both self-interested behavior and the ability to reproduce are only small parts of what drives the process of evolution in cultural and biological systems.
I think a lot of what is commonly seen as altruistic, is not actually altruistic, and that many people are not aware of their own motives for action. I believe true altruism is extremely rare.
No matter which way you slice it, it's the individuals who produce the most babies that make up the genetic code of any population. And the only thing that is truly common to people who produce babies is that they
want to produce babies. It isn't altruistic people who are more likely to reproduce, it's people who like having kids. And so our genetics are primarily comprised of a combination of genes that lead to that goal.
Contrary to your belief, it doesn't matter if the species propagates, it only matters that individuals within the species propagate. The species propagating is the corollary of individuals propagating, and so our evolutionary behaviour is aimed at the individual producing babies.
Like I said before, I would grant that altruism exists, but in practice altruism mostly only extends to family and in-tribes, and that's what accounts for your description of 'species wide genes' that ensure everyone makes it. The anthropological concept of the family gives individuals a supportive net that makes sure more kids get made.
To take that concept a little further, having children is an inherently selfish act in practise. If we were truly altruistic and giving we would be more likely to spend our energy helping people who need it rather than propagating our own genes. Because the world doesn't need more people, and a lot of people need help, but in reality.. nobody cares, because they have evolved to be baby-makers.
I don't mention that last part with any angst, because that's essentially what it means to be a living thing, but your idea that altruism is somehow baked into what we are, any further than extending to in-groups, is just not tenable to me. Maybe that's a grim picture, but that's reality, and until we as a species get better at understanding and respecting reality, we're not going to be able to make things much better