Do they really let those horses die of old age?
I can't speak for every country but in the UK, Olympic dressage horses retire from top level competition once they start to find the movements too hard work. At that point, there are a number of routes they may go: if they're a mare or stallion, they will be retired to stud and have foals. If they're a gelding, they may stop competing but carry on being ridden by up-and-coming riders who want to learn how certain movements should feel on a horse trained to the highest level, so they can in turn train their own horses. Sometimes they just become pleasure horses who are hacked around the lanes occasionally. Or they may retire completely to the field and never be ridden again.
It does most horses good to be kept in some form of light work for as long as they're physically and mentally capable of it, just as we do better for carrying on some form of exercise into later life. But if and when the time comes, when they're too stiff or old or ill or suffer an injury they can't recover from and are no longer enjoying life, they will be put to sleep.
A couple of examples: Charlotte Dujardin's triple gold medalling winning horse Valegro (known as Blueberry at home) retired a couple of years ago. He still lives on her yard, is hacked out to keep him supple and fit, and is occasionally used for lessons for other riders. There was a lovely clip on social media just before Covid hit of a young girl who visited the yard and got to ride him as a special treat, and he behaved impeccably with her.
Example 2: the horse Carl Hester rode at the 2004 Olympics, Escapado, went to a younger rider in the Netherlands a year or so later to teach him the ropes. When he was retired from competition altogether in 2011, he came home to Carl and lived a happy retirement. He was eventually PTS peacefully in 2019, aged 26.