Yes it would be correct to say that our body/genes don't cause ageing. Although our genes do try to delay it. These are genes associated with repair and genes regulating them. Mutations in these genes is associated with accelerated ageing and early onset of old age-associated diseases. These are genes involved in management of DNA damage repair, cellular senescence, inflammation and stem cell maintenance. At cellular and tissue level body has stem cells which is like a reservoir for all cells in that tissue. Stem cells provides precursor cells for all other cells in the tissue. This keeps the tissue young and healthy. Nevertheless, as we age then due to loss of stem cells or due to loss of their ability to differentiate (or lack of signals that activate stem cells or presence of inhibitory signalling) the tissue loses its ability to rejuvenate. An example of this is loss of melanocyte stem cell and/or their ability to differentiate leads to greying of hair.
If you look at the bigger picture, one of the major property of life is to keep the system away from equilibrium. Each cell and molecule in the body attempts to reach an equilibrium state and maximize the entropy. But active process in the body oppose these. But the nature eventually wins. With wear and tear accumulating over time these active processes malfunction. And when the equilibrium is reached cells and eventually we die.
Another way to look at ageing is loss of information. There is information in DNA for instance which is actively maintained by cells. However, cells lose their ability with age to maintain this information. Recent studies show that recording this information can reverse ageing.
So your second statement is also true. The mechanisms to avoid ageing are there, but they are not full-proof. Understanding these mechanism will help us slow down ageing (or even reverse it, hopefully).
I don't think that the third statement if true. We are not programmed to age and die. That's just law of nature (a default program) that everything just maximizes entropy and moves towards equilibrium. Life processes are about staying away from equilibrium and decreasing local entropy. Our genes/cells/and tissues tries their best to fight the nature. Nevertheless, the nature eventually wins.
Yes I once described life as anti-reality! Sometimes I even describe life as anti-life! In the sense that biological life basically opposes 'life' meaning reality, or nature, or entropy. But if, as physicists are telling us, entropy is a sure thing, then life is doomed to fail. So we're basically fighting a hopeless fight! I don't mean to sound nihilistic, cause I'm definitely anti-nihilist, but it's just an interesting way of looking at things.
The way I view biological bodies, is that we're already immortal, just not at the level we want. We are immortal at the gene level, not the person level. The genes use the body as a vehicle, and discard it as they see fit. But an old person can still produce a baby with young cheeks, young heart, free of cancer, etc., so in principle the ability to fix everything is there, never dying.
Loss of information is a rule of nature, like if you keep xeroxing a piece of printed A4 paper it will eventually just become a blank page or full of nonsense letters. But the constant pressure of natural selection gets rid of mistakes so the genes copying themselves don't get progressively worse at it. Trying their best to fight nature, as you say!