The claim is either of the two: - "Some people" might theoretically, out there, have diseases that are helped by eating animal products, so don't be judgmental - I have a disease that is helped by eating animal products
These include everything from the range of gastroparesis to various other disorders for which the link to diet is unexplored.
One common carnist claim is that their favorite food isn't vegan or there is simply not enough buy-in to make some product. The challenge is that there aren't enough dollars in such a market. If orders of magnitude more people were willing to buy veggie burgers then the variety would scale accordingly.
So unless it's medically necessary for you to eat non-vegan food, then by eating vegan and purchasing high-quality high-nutrient vegan products, you are contributing dollars into the economy towards the creation of more high-nutrient foods from plant-based sources. Your dollars are the strongest ballot you can cast in this area.
Instead, carnists are doing nothing different and then are surprised when high-quality vegan foods don't pop into existence from out of the blue, while lamenting that lab-grown meat isn't here yet and claiming defeat.
Gastroparesis is a terrible condition. In the worst cases, people will have entire months where they can literally ingest only some specific milk or vomit all the contents of their stomach up uncontrollably.
So there are some conditions which, indeed, require someone to eat non-vegan food. A breakup that was "really sad" isn't a reason to start torturing animals because it'll supposedly ease some imagined mental burden. In short, the number of people actually affected by serious conditions is vanishingly small, especially in proportion to the number of times this is used as an argument. If this isn't you, then you can't use this as a defense for your carnism, that the vegan isn't thinking of the poor unfortunate souls that need to eat animal products, so therefore that carte blanche is extended to you.
In addition to vague gestures towards self-diagnosed mental disorders, carnists will (in slightly better faith) claim that their diets are anti-inflammatory. Unfortunately, most of these diets include high-inflammation animal products and remove inflammation-fighting nutrients.
In addition, a lot of these claimed benefits are just the result of running elimination diets of essentially any type. But if it turns out you're gluten intolerant and you've narrowed your diet down to a carnivore diet, that doesn't mean that a vegan diet would de facto not solve those same problems.
It's also difficult to isolate how much of this supposed inability to go vegan is rooted in a preconceived notion that veganism is unhealthy because it's high-carb or causes inflammation even when such claims fly in the face of scientific evidence to the contrary. This seems like a likely explanation in at least some of the cases.