Recent advances in transformer-based architectures have shown promise in several machine learning tasks. In the audio domain, such architectures have been successfully utilised in the field of speech emotion recognition (SER). However, existing works have not evaluated the influence of model size and pre-training data on downstream performance, and have shown limited attention to generalisation, robustness, fairness, and efficiency. The present contribution conducts a thorough analysis of these aspects on several pre-trained variants of wav2vec 2.0 and HuBERT that we fine-tuned on the dimensions arousal, dominance, and valence of MSP-Podcast, while additionally using IEMOCAP and MOSI to test cross-corpus generalisation. To the best of our knowledge, we obtain the top performance for valence prediction without use of explicit linguistic information, with a concordance correlation coefficient (CCC) of. 638 on MSP-Podcast. Our investigations reveal that transformer-based architectures are more robust compared to a CNN-based baseline and fair with respect to gender groups, but not towards individual speakers. Finally, we show that their success on valence is based on implicit linguistic information, which explains why they perform on-par with recent multimodal approaches that explicitly utilise textual information. To make our findings reproducible, we release the best performing model to the community.