Got it working! Really simple solution that I somehow overlooked: I just hooked all the timelines to work off of Play From Start instead of Play.
The error was with the second and third timelines being set to Play; once they each finished running the timeline once, the timeslines were stuck at the end, and so the first timeline just kept looping over and over again as the only one playing from the beginning after the first runthrough of all three timelines.
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