The cost is not fixed. As you grow older, the insurance charges more to insure you. Yes, premium is fixed but the component that goes into the savings/investment/bonus (whatever you call it) is being reduced as you grow older. It doesn't matter if you buy it young or old.
The reason why you should get it when you are young is 1) you don't know when things will happen 2) when you're young and free of illness, insurance company will cover all. But if you're old and get sick, the insurance company may decide not to insure you or only give you partial coverage.