A domain (distribution) shift between training and test data often hinders the real-world performance of deep neural networks, necessitating unsupervised domain adaptation (UDA) to bridge this gap. Online source-free UDA has emerged as a solution for practical scenarios where access to source data is restricted and target data is received as a continuous stream. However, the open-world nature of many real-world applications additionally introduces category shifts meaning that the source and target label spaces may differ. Online source-free universal domain adaptation (SF-UniDA) addresses this challenge. Existing methods mainly rely on self-training with pseudo-labels, yet the relationship between pseudo-labeling and adaptation outcomes has not been studied yet. To bridge this gap, we conduct a systematic analysis through controlled experiments with simulated pseudo-labeling, offering valuable insights into pseudo-labeling for online SF-UniDA. Our findings reveal a substantial gap between the current state-of-the-art and the upper bound of adaptation achieved with perfect pseudo-labeling. Moreover, we show that a contrastive loss enables effective adaptation even with moderate pseudo-label accuracy, while a cross-entropy (CE) loss, though less robust to pseudo-label errors, achieves superior results when pseudo-labeling approaches perfection. Lastly, our findings indicate that pseudo-label accuracy is in general more crucial than quantity, suggesting that prioritizing fewer but high-confidence pseudo-labels is beneficial. Overall, our study highlights the critical role of pseudo-labeling in (online) SF-UniDA and provides actionable insights to drive future advancements in the field. Our code is available at https://github.com/pascalschlachter/PLAnalysis.