1-- I agree. 2-- I would hesitate to guess. However, Google gives 'with the click of a button' — About 18,600,000 results 'with a click of the button' — About 889,000 results The second phrase probably does not suggest only one button but more likely a specific button in the context (perhaps a user's manual)—which explains in part why #2 is less common than #1, which is a general idiomatic phrase (meaning, not precisely one click, but the general ease with which a task is accomplished).
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