Unfortunately, the reciprocal altruism breakthrough turned out to be illusory in the larger quest for the understanding of why humans are such a cooperative species. The problem is that it really works best for tiny groups of two, or very few people. Once the group becomes larger than 5–10, reciprocal altruism starts to break down, and it is certainly not the answer for lasting cooperation for any realistic group sizes, even in small-scale human societies (hundreds, or a few thousands of individuals)....and here's Peter J. Richerson and Robert Boyd (2005) in Not By Genes Alone:
Despite its many problems, theoretical work does make one fairly clear prediction that is revevant here: reciprocity can support cooperation in small groups, but not in larger ones. [...] Theoretical work suggests that this phenomenon will limit reciprocity to quite small groups, and while no good empirical data exist, it does fit with everyday experience. [...] We eventually stop inviting friends over to dinner if they never return our invitations; we become annoyed at our spouse if she does not take her turn watching the children; and we change auto mechanics if they repeatedly overcharge for repairs. But cooperation in larger groups cannot be based on the same principle.Both then use this "failure" of reciprocity to move on to hypotheses relating to group selection.
A prominent supporting paper for the idea that reciprocity only works in small groups appears to be a 1988 Richerson and Boyd paper - titled: "The evolution of reciprocity in sizable groups".
The conclusion of that paper says:
Reciprocity is likely to evolve only when reciprocating groups are quite small.The paper models cooperation within large groups by repeatedly sampling n individuals from the group and using an n-person Prisoner’s dilemma. However, that just doesn’t match up with how reciprocity actually works in large groups. Most reciprocal relationships – as the very word “reciprocal” might imply – involve two parties. That is the case where everyone seems to agree that reciprocity works. However, on the level of the individual, each person has a Dunbar’s number's worth of reciprocal relationships – throughout their social network. Then the cooperative network of each individual partly overlaps with the cooperative network of every other individual, creating cooperation that permeates the whole group.
The enlarged human brain allows more relationships to be tracked - and thus a larger reciprocative network to be maintained - than is possible for most other creatures.
Of course, reciprocity doesn't explain all cooperation between members of a group - but it is highly relevant to most within-group interactions. People don't engage in very many interactions with random group members and instead spend most of their time interacting within their social circles. There, they have one-to-one relationships, can remember the names and faces of people, can track their interaction history. It is under just these kinds of circumstances that reciprocity actually works.
So, reciprocity can - and does - help to establish cooperation within huge groups, provided that there are smaller networks within those groups - which, of course, there usually are. Reciprocity applies to cooperation within governments, companies, charities and other large-scale human organisations.
Reciprocity is a big and important pro-social force for humans - but it isn't all important. There are a range of other forces that also act to establish cooperation between humans in large groups - perhaps most notably virtue signalling.