the Nespresso support agent said that the machines should not be connected to a GFCI outlet
Where do they think Nespresso machines are used? Bedrooms? No, they're kitchen appliances and GFCIs are widely used in kitchens. GFCIs are code requirements and have been for ~20 years. If they are unable to make a GFCI-compatible machine, then their machine is unfit for purpose, and they have breached their implied warranty. Also, the things use water, for Pete's sake.
That said, machines do break. On a modern "mass produced, not expected to be repaired" machine, that means it is "at end-of-life".
This call with Nespresso was the first time I'd ever seen someone claim that GFCI outlets can potentially damage coffee machines.
LOL no. Such a claim is both nonsensical and inexcusable, and is certainly not coming from their technical department, nor has it been reviewed by their legal team, nor will you ever get them to put that in writing. Most likely it is because companies put tremendous pressure on customer service agents to have a high rate of good ratings, "close it in 1 call" and "avoid shipping replacement units". A GFCI-trip call is a worst-case scenario for an agent, and so they're going to say whatever they need to to get a satisfactory 1-call close. That is about their only option.
In a rare case, the GFCI might be end-of-life and not be working properly, but that is easily tested by plugging it into the next outlet over that is on a different GFCI, since kitchens generally have 2 independent circuits.
Now, other things can damage a coffee maker. Notably a "Lost Neutral", where the neutral lead from the power company breaks, causing neutral in the house to wander - the two 120V "halves of the service" still add up to 240V but they are lopsided, with one side higher than 120V. That is particularly hard on resistive heat appliances, since their heat increases by the square of voltage increase. Lost neutrals can be pernicious and difficult to detect, so it can go weeks or months without being noticed. It's not like service panels have voltmeters on them.