The problem is that this example is applying spoon theory to an almost completely different situation due to a misunderstanding of the theory as a result of continued dilution of its meaning to adapt the theory to non-chronic illness situations. In this way, it portrays spoon theory as a way to generally describe overburdening onesself with responsibilities who is otherwise functional, when spoon theory was originally meant to describe the significant effort in fundamental functioning. The original meaning, the meaning that applies to chronically ill people, is arguably entirely lost.
Disabled people's problems are not the same as healthy people's problems. Spoon theory was a way to describe exactly how it's not the same.
[x] people's problems are not the same as [y] people's problems
Maybe this helps to build understanding across x and y initially, but maintaining a hard and impassible empathic barrier between groups will eventually lead people to say, "Why should I care, every time I try to relate they say I'm not one of them?"
In this specific case, why is it such a bad thing to expand a metaphor for costs to build that shared empathy? Then we can all relate to each other better.
The metaphor isn't being expanded. It's being actively changed to no longer apply to the original situation. No shared empathy is being built because the situation being empathized with is not the original situation that the spoons theory was originally propgatated by.
Building empathy is the skill of identifying with situations that are not one's personal situations. Arguably, "expanding" a metaphor to suit one's personal situation is the opposite of fostering empathy.
EDIT: Towards "Why should I care, every time I try to relate they say I'm not one of them?" -> The spoon theory was not explicitly to be related to. It was a way to abstract a very real problem that has been broadly minimized into a finite situation in order to explain a phenomenon that a healthy person explicitly does not need to deal with and cannot relate to. It would be similar to me explaining a technical problem with metaphor; I'm not asking for you to relate to the problem. I'm asking you to understand the problem.
As for why should you care? Because this person is in a lot of pain and is dealing with a lot of difficulties that I'm not dealing with, and in fact, upwards of 40% (more maybe) of people in my country (USA) have a chronic/incurable condition. The scale of the problem means that developing a compassionate, understanding view will greatly benefit a significant portion of people one may meet.
Gatekeeping this intuitively universal spoon metaphor isn't going to make disabled people's lives better. A world where only disabled people get to use the spoon metaphor isn't going to have any more people feeling compassionate and empathic toward disabled people. The extension of an idea or metaphor to new situations doesn't devalue it; rather it shows that it is a powerful and useful idea. And before you make baseless claims, I also suffer from a chronic and incurable condition.
Disabled people's problems are not the same as healthy people's problems. Spoon theory was a way to describe exactly how it's not the same.