The "repairman" that moved the cord to an uprotected outlet needs replaced with a competent repairman or if the school maintenance man maybe should not be attempting this repair. Did anyone even open this machine up and check things out? I know you say it is old but it may only need less than $50 of parts or even just a little tape or other insulation over an exposed conductor to make it work, which cost much less than a new ice machine. If you don't want to look into it that is fine, your job then ended when you tell them there is something internally wrong with it. I have been in this boat myself a couple times - maybe not at a school but an appliance that trips GFCI's and owner or appliance repairman thinks something is wrong with GFCI. I have more than once had to show an appliance repairman what part needs replaced.
Exactly one of those times I would have looked harder into the appliance just to find the fault and show it to the appliance repairman, just because he had no business declaring the GFCI defective and to be able to go
Clamping a meter around both conductors of a circuit and getting a reading, no matter what that reading is, means some current is flowing outside the intended circuit, exactly how much current it was really was not too important in this case, I would still expect it to trip a GFCI if you are able to measure it at all.
Let's look at a few statistics:
1,000,000+ sq ft of buildings: about 20 campuses, some buildings built prior to 1900.
? dozen or so sports fields: lighting, scoreboards....
3 electricians (actually 2 electricians and a dufus).
Facilities all around the county. You can drive to each facility, spend 15 minutes, drive to the next ... takes about 10 hour to hit them all.
Relamp, change ballasts, repair broken switches, receptacles.
Install new circuits.
Wire computer labs for power and cat-5. Run conduit at each campus and co-ordinate with TELCO for high speed fiber.
Install and replace services from 100A/240V to 800A/208/120V and 480V 200A Y.
Rewire old buildings.
Wire room additions.
Bring power to trailer classrooms, install fire alarm and telephone systems in each.
Maintain and replace and install fire alarm systems, including door-lock and elevator interconnects.
Diagnose telephone key systems/PBXs prior to calling contractor for repair.
Design and install boiler control systems.
Maintain, with contractor assistance, 7200/12Kv pole line service on one campus w/failed pad mount transformer.
Equipment, including gas boilers, roof top units, gas water heaters, kitchen appliances .. when there was a problem the electricians were always sent first for diagnosis, then perhaps turn over to the plumbers.
Electricians sent to all "fire" incidents.
260 day contract, some years scads of overtime, some years, essentially none.
SO...
When I came upon a bad ice-maker (to be maintained by contractor) and noted that they had bypassed the GFCI protection, I didn't have the time to educate said contractor. The ice machine had the mechanical parts in the base of the machine, unenclosed, and no evident damage.