That’s odd that you measure 3v on the battery alone but nothing when it’s installed in the holder on the board. Even a mostly-dead battery should read something besides 0.000v. You usually only see that if there’s a short circuit somewhere on the board (which it doesn’t look like there is, from looking at the photos), but the battery would have been drained pretty quickly, and getting quite hot in the process, and definitely wouldn’t measure at 3v separately.
With the battery installed, measure the voltage at the two points shown by the red arrows in the attached picture. That’s the closest 2 points to the battery, so you should see something. If you’re not getting anything, make sure there’s not some tape or plastic anywhere on the battery or inside the holder on the Mayfly. Another thing to check is the shape of the battery. Is it perfectly flat on the negative (bottom) side, and if you set it down on a flat, level surface, can you tell that the battery is resting on only the negative terminal and the edges of the positive outer casing aren’t contacting the surface? What brand of battery is it? Some batteries can become misshapen or were manufactured wrong so they don’t make good contact with board in holders like these. Do you have an extra battery that you can try, or do you have more than one Mayfly board?