My guess is a seat switch or circuitry issue for the seat switch.

Ditto what Tom said-

Could it be that the switches are wired in 'series'?
So it could be and either-or situation.

"Could it be that the switches are wired in 'series'?
So it could be and either-or situation."

IF that were the case, the "good" switch would open the kill circuit.

Seat switch typically "trumps" the other kill switches.

I would think that the brake pedal switch was so that
it wouldn't turn over, (crank), unless someone was stepping on the brake .. . of course they would have to sit in the seat to step on the brake. I wonder what the OP meant when he said he 'tripped' the switch to keep the engine running. The seat switch is obviously a 'N.C.' switch, which would ground the coil if no one was in the seat,(or fell off the mower). I'm not sure what the
PTO switch is for. I'm sure there must be a 'truth table'
in the service manual, for diagnosing how all of these
switches work together.

Assuming a seat switch is NC can be a BAD assumption, if you don't have a schematic to confirm that. (I don't)

Sometimes, they deliver (or not) 12V to an Operator Presence Relay that switches the kill wire.

That's true, it is not good to assume.
But they way manufacturers are always
trying to build things cheaper, why use
a relay when a grounding switch alone,will
do the job.

To fool people that try to disable safety switches.