The huge claim is the present tense, "has the ability". It's not a huge claim to say that humanity has the potential to one day transcend money, but that wasn't the claim. Humanity has a long road before that's possible, it does not presently have the ability to continue to function if we just snapped our fingers tomorrow and eliminated money.
An "ability" is not a vague notion bolstered by historical curiosities. An "ability" involves a detailed, immediately actionable plan that can be implemented in the modern economic landscape without destroying crucial productivity.
Resources have to be allocated. People need to accept the resource allocation method in order to contribute their labor to do things that must be done. Money is an imperfect solution. Eliminating money leads to reinventing it (e.g. "energy credits"), reverting to less efficient models (e.g. barter), developing a central planning body that replaces wealth corruption with administrative corruption, or widespread social loafing where nothing gets done.
Without an actual plan of implementation that gains the trust of the workers, there is no "ability", merely aspiration.