Not finding anything on a Kohler 2700. Check the silver Kohler tag on the blower housing of the engine for the Model and spec number,
has the engine sat for awhile?
You're thinking of the (technically) called AntI-Afterfire valve, when 12volts is applied to it (you turn the key on) it retracts a plunger from the main jet. and when you turn the key off, power is killed and stops up the jet to prevent fuel from being drawn in the combustion chamber and out the exhaust making a POP, and raising the outside temp. somehow.
when you turn the key ON, not start, ON, listen for that solenoid to 'click'. however it may be retracting, but not able to hold open and kills the motor, MY preferred fix is to cut off the steel plunger rod flush with the solenoid threads. Engines ran without them for years, and yours will too. it will just act as a plug. And NO it will not cause the carb to flood the engine with fuel as some YouTube'rs falsely claim.