It’s not as much a computer as it is a safety device. It verifies all safety switches are in proper position before allowing the fuel cut-off to work (seat, pto and neutral switches).
I understand how a pull to kill works. I would have tried to fabricate something for the 790 but it was for sale and didn’t want it to be rigged up, bypassing safety switches.
Your 4400 will have a similar device or it wouldn’t be an electric solenoid.
Edit:
I looked up your 4400 and the thermistor part you spoke of and it serves a similar function in that it controls the voltage to the solenoid like my 790 module.
Here is a thread from another site discussing the problem.
https://tractorpoint.com/cgi-bin/tractor/pda_msgByThread.pl?parentnum=162657&bd=john-deere-tractor-review
Left cold because we aren’t supposed to link to other forums.