I am doing a remodel on a future home. On the initial walk thru we noticed the loud compressor noise from the frig. 2 months later frig died. Diagnosed by tech as bad GFCI or bad compressor inverter. I replaced GFCI and noticed much quieter compressor noise for 3 days so far. But my gut feeling is that the compressor inverter board is bad?
If the inverter board was faulty it could cause noise in the compressor motor, so your current solution seems to have possibly solved the problem.
Inverters generate pulse trains or sinusoids that are tuned to ramp up and ramp down and further adjust a compressor's motor "gently" - compared to an on/off relay switch.
Sometimes when we replace one thing a different problem comes to light. And sometimes a different previously hidden issue is unintentionally fixed, for instance a loose connection is redone, a loose mounting screw is tightened, etc..
But the GFCI had very likely nothing to do with the cause. The GFCI may have been nuisance tripping with the old board, and now with a new board and a new GFCI that problem has gone away.
Should you be bothered by GFCI nuisance tripping, then depending on then electrical code in your area you may be able to (or even have to) remove the fridge from the GFCI either by plugging the fridge in a different circuit or replacing the GFCI breaker/receptacle with a non-GFCI variant.
The electrical code must be applicable to your country, your state/province and even your municipality may have variations.
If removing a GFCI breaker, again check the code to determine which outlets must be protected (esp. plugs near sinks), and make sure they remain on a protected circuit, and get dedicated GFCI receptacles for those cases.
A GFCI does not make any effect on downstream devices, it is on affected by the type of load whether its resistive or capacitive/reactive and that is only in regarding to tripping characteristics.
Most likely your fridge/compressor is going out, when you turned off the fridge you reset the controller and hence reset the program operation of the thermostat in the fridge and hence it went quieter.