This is too long , and maybe also too important for a comment to Harper's answer:
Maybe there is a big difference in the statistics, but in Europe the most important/most frequent cases where a GFCI/RCD is saving lives and/or damages to the health (and also to property) is by preventing big fires.
F.e., the insurers say that most domestic fires start in kitchens (though by freezers relatively seldom) - the Greenfell Tower fire in London with ca. 70 deaths and dozens of destroyed apartments has been caused by a fridge-freezer.
A refrigerator should always be fed through an GFCI, since a fire in a fridge is most likely started by electric energy.
That energyflow will be interrupted by a GFCI as soon as the burnt cable isolation or a thin soot layer on the clearance surfaces will enable/increase current to the ground above the threshold.
The weakened/destroyed insulation in combination with heat increases the ground current right at the start of the fire, which will cut the electric power most likely early enough to avoid a catastrophe.
Dangers/deaths by electrocution is statistically less relevant in the context of GFCIs.
This thread reveals once more the important consequences in locations, where 5mA-GFCIs are mandatory, but 5mA-and only 5mA- GFCIs are allowed. From the opening post, it is not clear if the refrigerator is in a 5mA-only-and-no-higher-value- location, but the situation and reactions are typical.
Its about locations, where no higher ground currents are permitted.
Most likely, the optimal compromise is towards higher thresholds, which is realized in the big majority of all locations/countries on this planet, f.e. 30mA@230V.
Harper's answer does exactly reflect the problem: if 5mA is too sensitive, most people will decide or even recommend to use no GFCI at all - instead of using a 10 or 20mA GFCI (which is most probably not allowed thus not offered in that location) - or instead of buying more GFCI outlets to decrease the thus separated ground currents.
An isolation transformer will not at all help to fulfill the main GFCI- task - which is to prevent big fires.
This 5mA- or-nothing- policy is dangerous, it is somehow against its very own intentions - it can be back- firing, if users do not use or do bridge or eliminate an installed GFCI.
As for using a 12 years old fridge, it is a good idea to measure it's energy consumption - a replacement is not necessarily saving money or energy.
If there is enough space, some additional insulation can be attached to the sides thus reducing noise and energy consumption. That material has to comply with local rules and codes, thus simple polystyrene foam boards won't be allowed in many places, if a fire barrier like a metal coating is missing. (Thanks for this hint from the comments)
To check the doors' sealings, it helps to put a strong battery light into each compartment and to see where light is coming out if the door is closed. The place should be pitch dark.
As additional remarks and answer to comments:
At least in Europe, the priority with GFCIs/RCDs (there seems to be no difference in the intended meaning of both terminus technicus) is always to save lives - and avoiding fire can save lives, see the Greenfell fire.
But there seem to be different scientific examinations and interpretations, which threshold to choose.
In Europe, the related diagrams show that significant danger for live is assumed to start above 40mA.
It seems there are no data showing or indicating that 30mA is too high. Cases where deadly accidents with those 30mA-GFCIs would have been avoided with lower values are difficult if not impossible to find.
Another point is, that the time factor plays a big role, as it does with circuit breakers. A 5mA GFCI does not mean, the maximal ground current (through a human body) is limited to 5mA.
A solution would be a sealed/disabled switch at the GFCI, which is only enabled after n real trippings (excluding test button trippings) and decreases the sensitivity to f.e. 10mA or 7.5mA or what is found to be most efficient in that compromise in order to avoid by-passing or removing that GFCI - simply as it is much more work and inconvenient compared to toggle the enabled switch.
If a reader has statistics or documents which show the advantage of 5mA@120V instead of 10mA@120V, it would be appreciated.
10mA@120V equals 20mA@240V, which is still lower then 30mA@240V, but the human body's resistance is far from being linear.
About food safety: Are there statistics about harmful unnoticed power disruptions in refrigerators/freezers causing deaths or health issues?
It would be very simple and inexpensive to add an indicator in each fridge during manufacturing, but there seems to be no jurisdiction where this indicator is mandatory for household refrigerators - in the professional food industry the HACCP strategy is well established, so devices for upgrading refrigerators shouldn't be very expensive.
F.e., there are simple test stripes which change the color permanently if they are exposed to a certain temperature above a threshold.