Essentially every large mammal has intraspecies competition and assholery. So the theory that they developed in humans after humans had become a separate species is overly complicated and lacks significant supporting evidence. For extreme examples look at lions and chimpanzees, both of which have well-documented wars where they brutally kill each other.
Even putting that aside for a second, the claim of plentiful resources and sparse population has significant problems. First, sexual resources are always limited. Even when there is no other reason to fight, basically all large mammals and many other animals fight for sexual access. In order for a community to maintain a lack of sexual competition, it would have to actively punish sexual competition, enforcing promiscuity and making sure that everyone had access to sex whenever and with whomever they want (Brave New World style or something). Otherwise, individuals would have to strive to not be excluded (which is a form of sexual competition that might easily become violent). Moreover, any prohibition on sexual activity creates an incentive to cheat (which is a form of sexual competition that might easily become violent). All evidence I have seen has pointed to the origins of human civilization being strongly based on punishing sexual promiscuity and regulating sexual activity to an extreme.
Second, if resources really are that plentiful and population that sparse, this situation will last a couple generations at most. Population will explode and the community will expand. While initially groups may separate peacefully, they will still be in close proximity. And even if there are enough buffalo to go around, it's easier to steal one than kill your own. In short, if resource competition is not relevant in a society, the society will be able to grow quickly at which point it will quickly become relevant.
You're creating a false choice here by confusing Alpha with loner. A group with a strong dominant leader is not necessarily a small group. In fact, many people like following leaders like this, particularly when those leaders are adept at getting resources for the group (http://www.forbes.com/sites/daviddisalvo/2012/08/18/why-jerks-get-ahead/). Naturally, these leaders take more and the best for themselves, but if they are successful there is enough to go around and they will have many loyal followers. Human history is full of examples of this concept.
On the other hand, I am not aware of any examples of sexual promiscuity building a cohesive group. It tends to breed jealousy, backstabbing, and an unwillingness to contribute fully to the group. Isolated individuals may like this swinger type lifestyle, but most people react rather poorly to it. In early communist Russia there were attempts to create this type of community, but the results were rather disastrous.
Now, it may be that human nature has changed significantly in the last 10,000 years. You might even argue that the points I have raised here explain why humans are no longer the way they were 10,000 years ago. But this explanation has little evidence and seriously violates Occam's razor. It requires that 10,000 years ago humans were very different than other large mammals and that they evolved independently to be more similar. Without strong supporting evidence, that is an unlikely explanation.
As far as the biological evidence, sexual size ratios are going to have many contributing factors. The relatively small sexual size disparity in humans is generally seen as evidence that humans are mostly monogamous with some instances of polygamy. Even this conclusion is probably an oversimplification, but it's more consistent with other evidence than the conclusion that humans were group promiscuous in the recent past.
As far as testicular size, one would expect that the society proposed by "Sex at Dawn" would heavily favor men with the largest testicles, as the men who created the most sperm and could replenish their sperm the most quickly would be the most likely to father the children of the group. In point of fact, human testicles are rather small compared to many animals such as chimpanzees, for instance.