New work on thermalization in weakly perturbed quantum many-body systems has been published in Communications Physics of the Nature group. Proving thermalization from the unitary evolution of closed quantum systems is one of the oldest questions that is still only partially resolved. Efforts led to various versions of the eigenstate thermalization hypothesis (ETH), which implies thermalization under certain conditions. Whether the ETH holds in specific systems is however difficult to verify from the microscopic description of the system. In this work, we focus on thermalization under local Hamiltonians of low-entanglement initial states, which are operationally accessible in many natural physical settings, including schemes for testing thermalization in experiments and quantum simulators. We prove thermalization of these states under precise conditions that have operational significance. More specifically, motivated by arguments of unavoidable finite resolution, we define a random energy smoothing on local Hamiltonians that leads to local thermalization when the initial state has low entanglement. Finally we show that this transformation affects neither the Gibbs state locally nor, under generic smoothness conditions on the spectrum, the short-time dynamics.