How to fix earbud board with lost continuity?

Your question seems to contradict itself regarding the bottom right pad.

However, 2 of the pads, as you'll see, are for soldering the earphone lead to (let's call them A and D), and each of these is connected by printed circuit tracks to one of the remaining two (call them B and C, A connected to B and D connected to C). The two ends of the voice coil are soldered to B and C. You may have difficulty seeing this as the voicecoil is made up of very fine wire.

The only way you could have just one pad with no connectivity to any of the others is if one of the printed circuit tracks were broken. That doesn't seem likely, though it's just possible you broke the track in unsoldering the earphone wire. In that case you should be able to bridge the break with solder.

More likely is that the voicecoil wire is broken. This would show up by continuity between A and B and between C and D, but none between A or B and C or D. This would probably be very difficult to fix. If you have around 30 ohms between B and C your voicecoil is good.