Manufacturing systems envisioned for factories of the future will promote human-centricity for close collaboration in a shared working environment towards better overall productivity within the context of Industry 5.0. Robust and accurate recognition and prediction of human intentions are crucial to reliable and safe collaborative operations between humans and robots. For this purpose, this paper proposed a safety-aware human-centric collaborative assembly approach driven by function blocks, human action recognition for intention detection, and collision avoidance for safe robot control. Within the context, a deep learning-based recognition system is developed for high-accuracy human intention recognition and prediction, and an assembly feature-based approach driven by function blocks is presented for assembly execution and control. Thus, assembly features and human behaviours during assembly are formulated to support safe assembly actions. Skeleton-based human behaviours are defined as control inputs to an adaptive safety-aware scheme. The scheme includes collaborative and parallel mode-based pre-warning and obstacle avoidance approaches for a human-centric collaborative assembly system. The former is to monitor and regulate robot control modes when working in parallel with humans, and the latter uses a position-based approach to control robot actions by adaptively adjusting obstacle avoidance trajectories in a dynamic collaborative environment. The findings of this paper reveal the effectiveness of the developed system, as experimentally validated through an engine-assembly case study.