Test-Time Adaptation (TTA) has recently emerged as a promising strategy for tackling the problem of machine learning model robustness under distribution shifts by adapting the model during inference without access to any labels. Because of task difficulty, hyperparameters strongly influence the effectiveness of adaptation. However, the literature has provided little exploration into optimal hyperparameter selection. In this work, we tackle this problem by evaluating existing TTA methods using surrogate-based hp-selection strategies (which do not assume access to the test labels) to obtain a more realistic evaluation of their performance. We show that some of the recent state-of-the-art methods exhibit inferior performance compared to the previous algorithms when using our more realistic evaluation setup. Further, we show that forgetting is still a problem in TTA as the only method that is robust to hp-selection resets the model to the initial state at every step. We analyze different types of unsupervised selection strategies, and while they work reasonably well in most scenarios, the only strategies that work consistently well use some kind of supervision (either by a limited number of annotated test samples or by using pretraining data). Our findings underscore the need for further research with more rigorous benchmarking by explicitly stating model selection strategies, to facilitate which we open-source our code.