Utilizing air-traffic control (ATC) data for downstream natural-language processing tasks requires preprocessing steps. Key steps are the transcription of the data via automatic speech recognition (ASR) and speaker diarization, respectively speaker role detection (SRD) to divide the transcripts into pilot and air-traffic controller (ATCO) transcripts. While traditional approaches take on these tasks separately, we propose a transformer-based joint ASR-SRD system that solves both tasks jointly while relying on a standard ASR architecture. We compare this joint system against two cascaded approaches for ASR and SRD on multiple ATC datasets. Our study shows in which cases our joint system can outperform the two traditional approaches and in which cases the other architectures are preferable. We additionally evaluate how acoustic and lexical differences influence all architectures and show how to overcome them for our joint architecture.