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Federated learning (FL) is a promising strategy for performing privacy-preserving, distributed learning with a network of clients (i.e., edge devices). However, the data distribution among clients is often non-IID in nature, making efficient optimization difficult. To alleviate this issue, many FL algorithms focus on mitigating the effects of data heterogeneity across clients by introducing a variety of proximal terms, some incurring considerable compute and/or memory overheads, to restrain local updates with respect to the global model. Instead, we consider rethinking solutions to data heterogeneity in FL with a focus on local learning generality rather than proximal restriction. To this end, we first present a systematic study informed by second-order indicators to better understand algorithm effectiveness in FL. Interestingly, we find that standard regularization methods are surprisingly strong performers in mitigating data heterogeneity effects. Based on our findings, we further propose a simple and effective method, FedAlign, to overcome data heterogeneity and the pitfalls of previous methods. FedAlign achieves competitive accuracy with state-of-the-art FL methods across a variety of settings while minimizing computation and memory overhead. Code is available at https://github.com/mmendiet/FedAlign.