Federated averaging (FedAvg) is the most fundamental algorithm in Federated learning (FL). Previous theoretical results assert that FedAvg convergence and generalization degenerate under heterogeneous clients. However, recent empirical results show that FedAvg can perform well in many real-world heterogeneous tasks.These results reveal an inconsistency between FL theory and practice that is not fully explained. In this paper, we show that common heterogeneity measures contribute to this inconsistency based on rigorous convergence analysis.Furthermore, we introduce a new measure \textit{client consensus dynamics} and prove that \textit{FedAvg can effectively handle client heterogeneity when an appropriate aggregation strategy is used}. Building on this theoretical insight, we present a simple and effective FedAvg variant termed FedAWARE. Extensive experiments on three datasets and two modern neural network architectures demonstrate that FedAWARE ensures faster convergence and better generalization in heterogeneous client settings. Moreover, our results show that FedAWARE can significantly enhance the generalization performance of advanced FL algorithms when used as a plug-in module.