In summary, I dismiss the article as flawed (false and overly simplistic). No matter how noble (kind) your society, if the next society practices savagery and you are not prepared, your society will fail.

The level at which you pitch the concept of kindness is essential, as is the environment. To illustrate, many comments in this thread fail to hold for an example like the Mongols, or may argue that the Mongols wiping out civilisations was altruistic or kind. In a simplistic way, the Mongols survived through cooperation (as their society had some cruel rules that were definitely not kind), cruelty to outsiders and through theft. However, you could also contrast Chinggis Khan with Darius I of Persia who demonstrated far more kindness throughout his reign than Chinggis did. Both leaders in their own way were quite successful.

If you consider individuals, the dynamics can be quite different from those of nations, depending on the environment. If you limit the view to modern, first world societies, cooperation with ones peers is still essential and almost certainly more important than kindness. To illustrate, if I give everyone around me all of my money, that could be considered kind (think of a lottery winner). However, ultimately, such kindness (no matter how misguided) will usually lead to little or no benefit for the individual. Even if you compare two similarly cooperative people, the most competent cooperator will usually out perform a kinder cooperator. It will depend on the society as to whether an individual's empathy for those in need will add to the success of the individual and/or the community. For example, where a society has social welfare, an individual can point to the welfare system rather than directly contributing to the welfare of those in need.

Family dynamics are yet another level of complexity. Does a priority of kindness go to the family or the local community? What constitutes success of the family? Families that cooperate are more likely to succeed beyond those that are purely kind to each other. However, family cohesion is almost certainly dependent on both kindness and cooperation.

There are also work place environments, community group environments, local community environments, state and territory and country environments. In order to be successful, these environments require competency and cooperation. Kindness may or may not be required.

Regarding kindness.
In my observations of modern society (collectively), I observe that there are some examples that complicate concepts such as kindness. For example feminism* insists that domestic violence is something that only happens to females (through ignorance of statistics or for less savoury reasons). Ultimately, what this demonstrates is that kindness can be selectively applied and still be recognised as acceptable.

In summary, I don't see kindness as essential (at least not in all circumstances). Kindness can be selective (eg. Feminism's human rights failures with respect to domestic violence by representing only one gender) and, yet, such kindness can be seen as socially acceptable. Kindness can also be handed off to community and government groups in some circumstances. However, unlike kindness, cooperation is essential for individuals and groups.

* to be clear, feminism can not and should not be interchangeably used with female. Not all feminists are female and not all females are feminist. I think of feminism more like the KKK. Not all white people support the KKK and a criticism of the KKK is not a criticism of white people collectively.