We numerically study the effects of nonmagnetic impurities (vacancies) in the spin-S Heisenberg antiferromagnet on the kagome lattice. For a range of low but nonzero temperatures, and spin values that extend down to S =2 , we find that the magnetization response to an external magnetic field is consistent with the response of emergent "half-orphan" degrees of freedom that are expected to dominate the response of the corresponding classical magnet in a similar temperature range whenever there are two vacancies on the same triangle. Specifically, for all spin values we have considered (from S =1 /2 to 4), there is a large enhancement of the local susceptibility of the lone spin on such a triangle with two vacancies; in the presence of a uniform magnetic field h , this lone-spin behaves effectively as an almost free spin S in an effective field h /2 . Quite remarkably, in the zero-temperature limit, the ground state in the presence of a half-orphan has a nonzero total spin value SGS that shows a trend similar to S /2 when S ≥2 . These qualitative aspects of the response differ strikingly from the more conventional response of diluted samples without such half-orphan degrees of freedom. We discuss how these findings could be checked experimentally.