but if a mutation allowed an animal to extend its "prime" by several years, I don't see why this feature would disappear (except for overpopulation).
Well, that's it, for one thing. You have to assume that every population is at K (maximum capacity) or, if it isn't, eventually will be. Ergo there's
always competition for resources between individuals.
How does an animal with greater longetivity provide a survival advantage to its mature offspring? Especially if it's competing with resources with its offspring?
You've mentioned a few organisms with great longetivity, like trees, for instance. Trees can afford to live so long because they have a
lot of mechanisms for making sure they don't compete with their offspring for soil and light. Some, like fruiting trees, get other animals to cart their seeds far away. Others, like some pines, have seeds that won't sprout until there's a good chance the parent is already dead through forest fires.
Sea turtles swim for miles and miles to lay eggs, and abandon them.
There's no selection pressure for longetivity. In fact in the absence of a really good spreading mechanism, longetivity of the parent can select
against their offspring, because they compete for resources.