Garrett Hardin’s essay the Tragedy of the Commons wasn’t the first instance of the idea being written about by any means, not by a long shot, but it was one of the most important pieces for popularising it. Hardin doesn’t say anything explicitly racist, but he comes down pretty hard on the side of enforced population control and privatisation of everything. He even takes specific exception to the part of the UN’s universal declaration of human rights about the right to a family. While Hardin didn’t say anything like, “and we should control the population of black people first to make room for the whites,” (in the essay at least, the guy may well have been a massive raging racist elsewhere but I wouldn’t know), such Malthusian arguments are very often used to justify such beliefs.

Regarding the pro-capitalism side, this is something Hardin was pretty explicit about. One criticism of his essay is, as an example, that rather than enclosing sections of the commons in to individual parcels of private land, the community could share in the profits of the grazing animals instead, and then the incentive to abuse the commons is still handled. Perhaps this could still be seen as a sort of private property with shareholders if the community then winds up fending off a neighbouring community from using it, but I think for the purposes of one quick and short example of the limitations of Hardin’s thinking it works well enough.

You’re right that it’s pretty easy to find examples of it happening in real life. I think what we’re doing to the climate is probably the best possible example. However, Hardin and other writers typically don’t describe it as a thing that can happen, but a thing that will inevitably happen. In this case we do know that they’re wrong, ironically enough because of the commons that the term comes from. Hardin uses a broad variety of examples and doesn’t tie himself to the example of common grazing grounds, but the fact that such grazing grounds were successfully managed by communities for many centuries is something of a dent in the argument that humans will always follow the selfish incentive to abuse them.