Does your key-switch have position labels? What do they read left-to-right? My Perkins 4-108 has a key-switch labeled (left to right) "Off-On". But it has separate start and stop pushbuttons. The start button engages the starter and the stop button engages the stop solenoid (attached to the injector pump). Neither works unless the key-switch is in the "On" position but the engine will (once started) continue to run even if I return the key-switch to the "Off" position (but then my tach won't work). Will your engine continue to run with your key-switch in the center position? Will your tachometer register?
Is it possible that your three position key-switch engages the starter when turned to the far right and engages a stop solinoid when turned to the far left? In both instances the key-switch should return to the middle position when hand pressure is released. If the circumstances I've outlined above are true, then you may have a defective key-switch or the original key-switch has been replaced by one that does not return to the center position automatically.
This may be why " ... rev counter stays at about 700 rpm
even with the engine stopped. If I leave it like this it will drain the battery
overnight." The battery
drain may be caused by current
flow through the stop solenoid coil and have nothing to do with the tachometer at all.
If this isn't it, at least it's something to eliminate.