A relay coil, like a transformer, is a wire wound magnetic component. There are many different types of magnet wire wire insulation available-chosen for things like service temperature ( 80C to 240C ) breakdown voltage rating ( 1.5KV+ ), and manufacturing processes depending on design requirements. A simple solenoid coil is likely random wound on a bobbin with UL Class F (155C ) single layer insulated wire. The gauge and turns are design dependent.
Just as nothing is perfect the same applies to mag wire insulation. Its typically very thin ( .001 in ) which will vary ( tolerance ) and even have tiny pinholes. Winding operations can cause nicks and scratches. It is commonly assumed that the like likelyhood of any defects lining up to cause turn to turn faults is remote. Properly designed manufactured, and operated a coil will last for 20+ years.
However, a solenoid coil is inductive. When power is removed from an energized coil a high voltage discharge is created. Where that energy discharges can be harmless or can damage mag wire insulation. Repeated discharges within a coil can result in turn to turn shorts and failure of the coil. A reverse biased diode is often placed across DC relay power terminals to snub that arc but It has zero effect on the switching contacts.
So just replace it with a cheaper model/kind that has the same specs. Who said you had to have the exact same one as long as the specs are the same ? Was it $288.00 for the replacement....Frack that! You could replace it ten times over for years for that price. just like OEM products are always more expensive than after market and are probably a little better but way over priced.
A little history as far as my experiences.
Coming to N.A. in the 70's and from a European background, I was accustomed to the use of DC solenoids in industrial control, where failure rates where considered quite low.
Once getting in to N.A. Industrial control, I found a prevalence for 120vac was the norm for hydraulic/pneumatic valves and solenoids and failure rates seemed high.
One of the companies I did work for converting relay logic to PLC control, I specified DC solenoids, I got some push back from the local suppliers, as DC solenoids were only just coming into popularity due to useage on mobile hydraulic equipment etc.
One reply I got against going DC was, "we have always done it this way"!
I took a look at the companies maintenance records and apart from the stock of spare AC solenoid coils, was the amount of down time attributed to hydraulic valve coil failure.
It was not helped, BTW, by maintenance personnel when trouble shooting, forcing the armature of a energized valve over in order to 'diagnose' the problem and in the process. destroying the coil.
This would not happen with the DC version.!
Hey so I tried the rectified 12v off a small transformer into a retro fit. No go. Not sure why. I installed a 220vac solenoid off Aliexpress cost $8. Was a little tricky getting the connection right but it works great! Job done until the next part craps out on my 10yr old fridge ha