Detecting collaborative and problem-solving behaviours from digital traces to interpret students' collaborative problem solving (CPS) competency is a long-term goal in the Artificial Intelligence in Education (AIEd) field. Although multimodal data and advanced models are argued to have the potential to detect complex CPS behaviours, empirical evidence on their value remains limited with some contrasting evidence. In this study, we investigated the potential of multimodal data to improve model performance in diagnosing 78 secondary school students' CPS subskills and indicators in authentic educational settings. In particular, text embeddings from verbal data and acoustic embeddings from audio data were used in a multimodal classification model for CPS diagnosis. Both unimodal and multimodal transformer-based models outperformed traditional models in detecting CPS classes. Although the inclusion of multimodality did not improve the performance of traditional unimodal models, its integration into transformer-based models demonstrated improved performance for diagnosing social-cognitive CPS classes compared to unimodal transformer-based models. Based on the results, the paper argues that multimodality and the selection of a particular modelling technique should not be taken for granted to achieve the best performance in the automated detection of every CPS subskill and indicator. Rather, their value is limited to certain types of CPS indicators, affected by the complexity of the labels, and dependent on the composition of indicators in the dataset. We conclude the paper by discussing the required nuance when considering the value of LLMs and multimodality in automated CPS diagnosis, highlighting the need for human-AI complementarity, and proposing the exploration of relevant model architectures and techniques to improve CPS diagnosis in authentic educational contexts.