Pictures of the circuit board won’t likely be of any use. A schematic or pictures of the inside of the original pedal might help. As IC Racer pointed out, sometimes the ground is necessary for operation. Though, usually when it it, they don’t route it directly to the chassis like in the picture. However, there are two other wires grounded there as well, which is odd. It could be a part of a Star ground setup, which is uncommon for something like a timer that doesn’t require an ultra low noise circuit to function properly. It’s far more common in hifi audio equipment and such, where people want to remove every tiny bit of static from the signal. Or, it could have been designed by someone who knew electronics, but didn’t know common practices, so they just did it in a nonstandard way.
If you can find a schematic or pics of the inside of the original footswitch, we’ll know for sure. We need a reference point to know what the switch is supposed to do to find out what works and what doesn’t.