Precise note-onset timing is crucial for training automatic music transcription (AMT) systems, yet onset-accurate annotations for real-world recordings are scarce. Sequence-level score–audio alignment methods such as dynamic time warping provide only coarse correspondence, making an onset-level refinement step necessary. This refinement step, known as snapping, adjusts aligned score onsets using peaks in a neural onset posteriorgram and often determines whether weakly aligned score--audio pairs become usable training data at all. Despite its practical importance, snapping is typically treated as a simple post-processing heuristic and implemented with greedy local decisions. We present a systematic analysis of snapping strategies for training instrument-agnostic transcribers, demonstrating that snapping is essential for learning from weakly aligned data. Building on this, we formulate snapping as a per-pitch assignment problem and solve it via bipartite graph matching, yielding context-aware onset decisions under overlapping refinement windows and uncertain initial alignments. Extensive cross-dataset experiments across piano, chamber, and orchestral recordings show improved onset alignment and transcription accuracy over greedy snapping, with gains increasing for wider snapping windows and coarser initial alignments.