I got about halfway through that podcast. Some good points to be sure, but also a depressing listen. The fact that the host, and I'm assuming the author, are both atheists is very telling. But having said that, if even if I was an atheist I would not want immortality. One of the author's points I agreed with was when she said she wasn't sure if humans were psychologically built for immortality and continued living in perpetuity might drive people insane. I also found interesting that her father, apparently a very religious man, was afraid of dying right up to the end. (Part of her take on that was different than mine, btw.) In my own family, I know my father was totally prepared to die. In our very last conversation before going in for heart surgery, he told me that the surgery "would either kill me or make me feel better; I'm fine either way." And I knew he meant it. He never came out of the anathesia and passed a few weeks later. My mother had already passed and I think he was lonely in a way that couldn't be fixed. He was a very devout man and I'm sure he believed heaven existed and that he would be there. In the case of my FIL, who passed last February (two weeks shy of his 98th birthday!), I believe his only regret in passing was leaving his wife of 70+ years alone. Otherwise, he too was quite prepared.
In the Lord of the Rings, Tolkien explored the concept of the immortality of the elves versus the mortality of men. The elves refer to mortality as a gift from Eru Iluvatar - who is the god figure of Middle Earth - as they have become keenly aware of the drawbacks of living forever. The mortal men of course, are not capable of understanding how exactly it's a gift. The reader on the other hand, totally grasps it.