It does not. Harm is not death. Death is death. Immortality is the inability to die. Thats it. It is not the inability to be harmed, afflicted, debilitated, pained, or anything else. Its the inability to die.
You seem to recognize that just immortality without any sort of requisite secondary powers would suck really hard, so im not sure why youre pushing back on the idea that a lot of people wouldnt want immortality.