I do not see that. When assisted suicide is allowed only in cases of impending, inevitable death I don't see it as choosing to die, but rather choosing how to die. I see it as akin to a condemned prisoner asking for lethal injection as opposed to the electric chair. The prisoner didn't, by way of choosing the method of his death, choose to die. The death sentence has already been passed down. The terminal cancer patient, like the condemned prisoner, did not choose to die. But death is fast approaching, and options are available as to how that death occurs. So, from the perspective of society and of government, we will not help you if you choose to die. But, if you are dying, we will help you choose how it occurs
As Athelstan notes, everyone dies, so death is inevitable for everyone. Therefore every suicide could be viewed as a choice of how to die.
This is particularly important to realize because in many cases where we are asked to allow for assisted suicide, the patient may have a terminal disease, but one with manageable symptoms and one which may not kill them for years (though of course it is impossible to say
exactly how long any patient may live). Generally we aren't discussing situations where there is a risk of the patient dying the next day.
Another important note here is that how long someone can live is often dependent on the amount of help that they receive from others. As an analogy (and as we've learned to be careful about analogies, I will spell out exactly what I think is relevant about this analogy), consider the following: Suppose we came across a car accident, and one of the people involved was severely injured, but conscious. He tells us "this pain is unbearable, and so I wish I could die" or "I can't feel my legs and I'd rather die than live without being to walk" or in some other way expresses a desire to die because of his condition. Do we call an ambulance (or at the very least try to convince him that he should not give up on life) or do we help him kill himself?
Now here is what I think is relevant about this example: the man really is dying and he really does want to die. There is no guarantee that we will be able to save him if he does receive medical attention, because his injuries are severe. If he does survive, he may be permanently crippled. So in many sense this is a terminal patient, with only two differences that I can see between him and a terminally ill patient: the first that he is injured and not ill (but I don't think that the
way that someone gets close to death has entered into our discussion of suicide) and the second is that he is probably actually closer to death than a terminally ill patient (which by your reasoning would seem to give him
more justification to demand suicide).
Or consider someone with a mental disease that leads to very strong suicidal compulsions. In fact, we can say that these compulsions are frequent enough and strong enough that left unassisted and untreated, that person will commit suicide. Then in a sense we could say that that person is dying of the disease, since given enough time it will kill him. Would we tell this person that we understand why he wants to commit suicide, or would we offer him treatment?
All these situations are very different from your example of a prisoner on death row, because the prisoner on death row has literally been sentenced to death (and after a point, he will even now the day and perhaps even the hour of his death). It is not a question of "given enough time, he will die because of an execution" but rather that we are intentionally arranging so that will happen (even saving his life should he die before the appointed execution!) Because of all of this his death is much more certain than a terminally ill patient. A terminally ill patient can at least in principle be saved, but (if we do not change our minds and we have good security) the death row prisoner cannot.
Even with those problems, your analogy would be a little more accurate if you spoke about a death row prisoner who found a way to hang himself in his cell rather than wait for his execution. But I think that even this would make your analogy less compelling.