Their reproductive lives are so drawn out that surely there's intense pressure to stay healthier for longer. They reach sexual maturity as late or later than us, gestate for at least twice the duration, care for their young for a long time, have relatively few offspring, and can keep going, sometimes, till their sixties. All that extra time to raise few children means evolution must select those who can keep ahead of attrition, and it's a marathon not a sprint.
Probably a lot of elephants die to infections because they don't have antibiotics, or famine or poaching or the flu or a lot of other causes that we reduced.
It's so high because we eradicated so many other things that used to kill us off early.
Looking at the leading causes of death in the UK[1] they're: dementia, heart disease, strokes, lung cancer, flu. If you lowered any of those then the rates of the others would go up.