Those switches should have 4 wires going to them
2 are white & 2 are black
The white ones are the cranking circuit so should be closed when the plunger is in . They are POWER WIRES
The black ones are the kill circuits and should be open when the plunger is in and closed when the plunger is out.
They are a ground circuit & go to the magneto to stop it making a spark.
If they accidentially get battery power then they will kill the magneto .
Now get some shortish pieces of wire and put a male spade on each end
Unplug the switches, bridge the white wires and leave the blacks open
Unplug the reverse switch which will be on then end of the wires that branch off the seat switch
The mower should work as normal
If so plug the switches back one at a time till it plays up again and you have found the guilty party
If it still wont run then the seat switch, or the wires going to or from it are the problem .
I have a lot of problems caused by the new anti removal set switches to the point that I have been retro fitting the old switches off mowers in the graveyard.
Some what ironic that a part that very rarely caused problems got replaced with a part that regularly causes problems because people incorrectly blamed it for causing problems.