We study the validity of the eigenstate thermalization hypothesis (ETH) and its role for the occurrence of initial-state independent (ISI) equilibration in closed quantum many-body systems. Using the concept of dynamical typicality, we present an extensive numerical analysis of energy exchange in integrable and nonintegrable spin-1/2 systems of large size outside the range of exact diagonalization. In the case of nonintegrable systems, our finite-size scaling shows that the ETH becomes valid in the thermodynamic limit and can serve as the underlying mechanism for ISI equilibration. In the case of integrable systems, however, indication of ISI equilibration has been observed despite the violation of the ETH. We establish a connection between this observation and the need of choosing a proper parameter within the ETH.