They sell a lot of ignition switches because every time somebody has an electrical problem they think it is the ignition switch, when in reality switch failure is rare.
Followed closely by removing the seat switch despite the fact that it is not in either of the starting circuits.
Now back to your problem
One of the safety cut outs is working too good
This will be the brake + seat switch
PTO + Seat switch
If it is a zero turn add
Brake & lap bars
And if you have a reverse cut out.
Reverse & PTO switch.
All of the safety switches work in pairs.
So individually they can be fine with a problem in the wire between them
'And to make matters harder, all of the safety cuts outs are ground circuits, so finding a short to ground of a ground wire is a PIA.
Depending upon how old your mower is, some of these switches will have multiple contacts and some will be power wires ( cranking circuit ) so be careful, if you cross over a ground & power wire then it is bye bye coils and you already know how expensive they are.
THE service manual for your mower is also available on line.
If you can not find it, start another new thread titled looking for a manual for a ???????????????.
There are some really internet clever people on here who can ferret this stuff out really well ( better them me in any case )