Immortality is a recurring theme in 3 of the stories but based on the others and what I have read he is not trapped in a loop as some writers are.

Not that you can't write a ton of good stuff using a recurring theme. Stephen King has done well with that.

I have read a number of his short stories (none of which stand out)and two of his novels. At Winters End and it's sequel, The New Springtime.

Humanity has abandoned Earth after an extinction level impact event. I don't recall whether they transcended or just left in spaceships. Humanity was highly evolved at the time and had already spread through the galaxy. Humanity never reappears in the two novels but they are ever present as a source of the Simian's religion and folklore.

When they left they established a bunker far underground for Simians that they believed were evolving as then next intelligent species.

Winter's End refers to the time when the Earth starts to thaw from a 100,000 year ice age that had engulfed the planet after the impacts.

The story is about the Simians leaving the shelter and reclaiming the Earth. They are still culturally backward, similar to early homo sapiens but they are somewhat aided by simple tech from the shelter. They later uncover more human technology that gives them a big jump start.

The New Springtime sequel is about their spreading out shedding their superstitions and dealing with an intelligent insect species that has also survived.

I also recall the TWZ episode mentioned but I don't recall connecting it with Silverberg at the time.

The novels you mentioned sound interesting and I would include them on my very long list of things I would like to do but might never get around to. :)