I will forgive you for this because when you looked in the back of your parts book that you got with the mower or downloaded because you lost it Scag do not bother to show the switch positions
If they did you would have seen that in your switch there are 3 individual sets of connectors
1 turns the PTO on & off
1 completes the cranking circuit
1 activates the engine cut out
So just because 1 pair are working, does not mean that all 3 sets are working
The connector works flat to flat on the pins
So in the off position 1 pair should have continuity and the other 2 should not
This one pair is the cranking circuit which should be the light blue to red
If you jump those positions the engine should crank when you turn the key to start ( providing the rest of the system is in the right positions )
IF not then the module is bad $$$$$
But before opening your wallet, pull the module plug saturate it with whatever you have WD 40 - Brake Cleaner - contact cleaner then connect & disconnect it several times as the plug is prone to corrosion.