Where does he talk about immortality? When he talk of "eternity", I think he's talking about mankind, not himself; same for the inevitable end to life - he's talking about life in general.
I suspect that the whole SETI business derives from the same bugs in our evolved theory of mind which have also given us god beliefs, the Turing Test, conversations with Siri apps and other delusions.We have no evidence that minds similar to what humans display exist anywhere else in the universe.
We also shouldn't get carried away with our lucky guess about the existence of exoplanets. Exoplanets do not imply ET. Until about 20 years ago when exoplanets started to become observable, even "skeptics" believed in their existence even though they met their own criteria for extraordinary claims, as in Carl Sagan's essay, "The Dragon in my Garage." We now call the Dragon ET instead of exoplanets.
To flip this around:
I suspect that the whole we're-alone-in-the-universe business derives from the same bugs in our evolved theory of mind which have also given us geocentrism, "Man is the measure of all things," literal Adam-and-Eve creationism, humans-with-bumpy-foreheads in Star Trek, Teilhard de Chardin's "Omega Point," faith that AGI will be human-friendly by default because AGI is so smart and humans are obviously great, and other delusions. We have no evidence that Earth and humanity are special to the point of utter uniqueness, that we are the only intelligences in the universe.
(My point here is not to argue that "the whole SETI business" is a good idea. It is to illustrate that the above style of argument doesn't work very well either way.)
I gather that you accept the Copernican Principle, but I've noticed that cosmologists don't seem to agree upon that. The Copernican Principle would seem to imply that all observers in the universe, regardless of their respective positions in its history, would have the same ability to infer the universe's origins. Yet I've also read arguments that we live in a privileged time in the universe's history when we can still observe distant galaxies, measure their red shifts, and therefore can infer the Big Bang. Astronomers living X billions of years from now would find themselves in isolated galaxies after all the other galaxies have passed beyond their light cones, so that they would lack the ability to infer the Big Bang, at least in the way our astronomers have. I don't know what model of origins they might come up with, but it probably wouldn't resemble the one we know.
In other words, according to this argument, either one significant violation of the Copernican Principle exists so that we do live in a privileged position in the universe's history, analogous to geocentrism (how convenient for academic cosmologists in their status games); or else the universe's inferable origins change retroactively depending on the time in the universe's history astronomers make observations of the cosmos.
Since the cosmologists themselves apparently haven't made up their minds about the Copernican Principle when it comes to what looks like motivated reasoning on their part, I think the principle deserves skepticism when you want to apply it to the existence of ET's.
It's interesting to see someone with actual political power (with balance of power in the Senate and being part of the Lower House coalition) talking about the Great Filter.