Well in a way you can say that ageing is indeed the cause of most deaths. One of popular theory of ageing however is accumulation of molecular, cellular and tissue damage. Repairing the damage is of utmost importance at any level. Both failing to repair the damage and going on an overdrive can prove detrimental. At molecular level the damage is mutations accumulated in somatic cells with age. And DNA repair pathways often try to fix them. But take example of choosing to repair vs choosing to initiate senescence or apoptosis. There needs to be fine balance maintained between these pathways. Overdrive of Non-homologus end joining can cause further DNA damage, excess of senescence however may induce senescence associated secreted proteins (SASPs) which increase inflammation and promote tissue damage.
I like the example of cardiovascular diseases because it exemplifies both importance if ageing and fibrosis. First myocardial infarction (heart attack) maybe induced by underlying atherosclerosis (associated with vascular fibrosis) or due to injury to heart cells due to glycemic injury (in diabetics) or underlying low-grade inflammation owing to ageing and stress. The lack of oxygen can then cause ischemic injury, which the heart would try to repair after the recovery from the attack. The healing process often leads to a scar, which hinders proper functioning of the heart. The second best example is of solid tumors, a mass of cancerous cells (often more likely in old age) induce a desmoplastic reaction, causing fibrosis, which hinders function of the tissue where the tumour is. On the other hand, fibrotic tissues are highly prone to development of cancers. In fact prostate cancers before spreading to bone remodels the bone matrix before establishing secondary site of the tumor. So while age can be attributed as the most obvious cause of death, the tissue hardening due to accumulated damage or due to age associated disease is what causes tissue dysfunction. Fibrosis is a condition or state of tissue rather than being a disease itself.
Now of course there are deaths that happen which doesn't involve fibrosis (about 55%) but other tissue conditions. These include failure to repair damage (chronic wounds for example), traumatic injuries (accidents and suicides), cytokine storm, sepsis, haemorrhages or other acute failures.