The no continuity is expected. You would only get a beep if the button is pressed (the press happens when both sides are connected by a conductor, which in the case here is the carbon on the button membranes). It could be the contacts are just dirty and cleaning them with a q-tip and ipa is good enough. I would also lightly clean the membranes as well to make sure there is no dust or residue on them (the carbon on the button will always come off on the qtip, so just lightly clean that part).
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I have already cleaned the contacts and the membranes. How would I go about tracing where the problem lies? I know there's no continuity when the button isn't pressed, but nothing happens when I bridge any of the buttons, either with the membrane or otherwise, with the exception of the start button, which then seems to trigger all the buttons continuously (judging by the leds).
It's really odd you wouldn't get continuity even using metal across the buttons. That seems like a physical issue with those contacts.
Do you know how I can proceed tracing where the issue lies? Power is applied, as I can get the leds to light up.