I think this "would begin again" indicates a past habitual action along with other sentences' "could" and different verbs, not past willingness, right?
32)In an experiment, a monkey was subjected to a mild but uncomfortable eletric shock every twenty seconds. If the monkey was to avoid the shock, he had to push a red button to reset the clock. The twenty-second period would begin again. Thus, the monkey could stop the electrical shocks if he could locate the button and press it before the twenty-second period had passed. If he passed the button at the correct time, the monkey could avoid all the shocks...