What both you & Super are missing is the fuel pump should pump up the system when the switch is turned ON way before he hits the Start Position
If you own a fuel injected car you have the same system.
When the key gets turned to the on position the fuel pump kicks in to pressurise the system.
This is what is not happening.
The pump relay is not firing up the pump unless the battery charger is on a very high setting and would be over voltage.
Depending upon which system he has either the ECU is duff, or the fuel pressure sensor ( if it has one ) is not sending a low pressure signal to the ECU so it fires up the pump.
So from here on Mowerman556 will have to go to the Kohler web site & download the service manual for his mower.
As he mentioned a "plug with about 40 pins" I will assume he has the 32 pin Bosch controller on his mower.
We have gone through all the easy stuff, now it will be a case of running the fuel pump control circuit diagnostics,
AS he has not mentioned an error code this would tend towards a problem with the ECU itself .
We do not know which engine & which injection system Mowerman has and even if he did there is very little I can do from here on.
The pump is in fact working and at some point the controls for the pump are working but the start up cycle is not unless the system gets an overwhelming voltage.
This normally would suggest a bad connection somewhere but he has done all of the jumping from the battery that he can so now we are left with the ECU & the pump relay, if the later is not part of the ECU or possibly the Fuel pressure sensor ( should it have one ).
According to how I interpret the manuals I have the unit should be showing a service warning if the pump , relay or pressure controller are bad and he is not getting one.
All that can be done now is to see if the ECU is putting out an error signal by hooking up a test lamp or reader to it.
The problem with all this stuff is the sensors used put out very low voltage signals and in many cases it is only a fraction of 1 V so any corrosion on any wire can cause a fault as could any terminal where all but a few wires have broken.
Mowerman does not have the gear to read the outputs ( neither do I ) .
I keep some old dial voltmeters for making these readings ( best I can ) and bought a way too expensive LED test light that has never proved to be reliable.
Good observations and info. This is an a very interesting problem and troubleshooting thread. My experience with similar issues are with my '06 motorcycle that is fuel injected. While it is these are different machines, they shar a general design. The issues with my bike were traced to the main ground from the battery to the first connector(badly corroded). Adding extra grounds can be a big help, in my opinion, but doesn't help in this case. As you mentioned about a bad connection, that's worth exploring. It will take a patient and determined troubleshooter to track this problem down. It wouldn't surprise me that the root of the problem is very simple. I hope the OP will share the final solution.