Unsupervised source-free domain adaptation methods aim to train a model to be used in the target domain utilizing the pretrained source-domain model and unlabeled target-domain data, where the source data may not be accessible due to intellectual property or privacy issues. These methods frequently utilize self-training with pseudo-labeling thresholded by prediction confidence. In a source-free scenario, only supervision comes from target data, and thresholding limits the contribution of the self-training. In this study, we utilize self-training with a mean-teacher approach. The student network is trained with all predictions of the teacher network. Instead of thresholding the predictions, the gradients calculated from the pseudo-labels are weighted based on the reliability of the teacher's predictions. We propose a novel method that uses proxy-based metric learning to estimate reliability. We train a metric network on the encoder features of the teacher network. Since the teacher is updated with the moving average, the encoder feature space is slowly changing. Therefore, the metric network can be updated in training time, which enables end-to-end training. We also propose a metric-based online ClassMix method to augment the input of the student network where the patches to be mixed are decided based on the metric reliability. We evaluated our method in synthetic-to-real and cross-city scenarios. The benchmarks show that our method significantly outperforms the existing state-of-the-art methods.