Every pushbutton has a limited number of times you can press it. A close door button is usually pressed many times by impatient passengers. So it will wear out more quickly than any other button.
1) Buttons themselves have on the order of hundreds of thousands to millions of presses expected before any form of failure could begin to be expected. OPs story of a new lift failing would not even come close with a person standing there 24/7 mashing the button
2) A button failing would be just that: a failed button. Not an entirely failed control system. One would assume the button to be a Normally-Open activation, such that continuity in the circuit is achieved when the button press occurs so a failed button would look to be like a button that hasn't been presssed anyway. I'd be happy to be corrected on this, but a Normally Closed button is often only really used for safety sake where a break in the line, loss of voltage, or a button press are all treated equally by the control system.