The problem is that death isn't caused by telomere shortening. That might kill *some* cells (or even a lot of cells), but it's not something that will strike all of our trillions of cells at the same time. So, while it may be responsible for slow degradation due to age, it's not really responsible for death.
The thing is that our cells are specialized. REALLY specialized. They're specialized to the point where they can't really survive on their own anymore. Even if they could, most of them are trapped deep within our bodies, and isolated from any source of nutrients. That's why we have systems in place to feed those cells. Our digestive system extracts nutrients from food, and our circulatory system brings those nutrients and gasses to cells in order to keep them alive. If our circulatory system breaks down, our body is no longer able to maintain those cells, and we die.
The brain is even more specialized. Neurons are very picky when it comes to metabolism. They require oxygen (they can't carry out anaerobic respiration, like most cells can over the short term), and they require a constant supply of glucose. That means that once circulation breaks down, neurons are the first to die. After that, it's sort of a vicious cycle, as the brain can no longer regulate the rest of the body, hastening its death. On top of this, who and what we are is encoded into the physical structure and activity of the brain. Once the cells die, that structure and activity break down, and the self ceases to be.
So apoptosis and telomere shortening, as well as accumulation of damage and mutations, can lead to degradation, it's not really the mechanistic cause of death. It may damage the circulatory system (or any other vital system) to the point where it can no longer function, the true cause of death is the breakdown of that system itself. An analogy would be my house suddenly collapsed and killed me. The termites in the woodwork might be what sets the whole collapse in motion, but in the end, it's the falling support beam that bonks me on the head that does me in - it's the collapse itself that kills me, not the conditions leading up to the collapse.