I'm sure there was also the gradual realization, conscious or not, that space is also one of the very few ventures where you get out of it what you put into it. Spend $10 billion on SpaceX, and we'll have ships headed to Mars to establish the first infrastructure of human civilization there. By contrast you can't just buy your way to a good education system, or spend away social problems. Spend it on most other things and you're going to see a relatively negligible positive impact. Spend it on our favorite way to waste money (and with orders of magnitude larger sums), geopolitics, and you'll likely see an overall negative impact.
I don't really see NASA playing a meaningful role in the future of space. Their role in the present, outside of funding, is already quite negligible. They lack consistent leadership/direction, and are driven by congress - who is driven by graft. The SLS (NASA's new rocket - made by Boeing/Lockheed) alone has hoovered up tens of billions of dollars, to one day hope to construct a rocket that already isn't competitive against a Falcon Heavy. At the same time SpaceX is constructing a ship which will revolutionize space, again. China has already designed, built, and launched a new space station into space (as ours continues to deteriorate), and so on. The systems and motivations which drove NASA to epic success in the 60s simply no longer exist in our public sector.
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The other things sound reasonable, but seem largely driven by a lack of knowledge - which I am not directing at you, but the authors in the podcast you linked. For instance you don't need to live deep underground to protect from radiation. One of the best shields against radiation is hydrogen. So one of the ways we'll likely protect against space weather on the way to Mars is by using the water storage on the ship as a shield. On Mars igloos, as odd as it sound, would make beasts of cheaply constructible radiation proofed housing, for but one example. You'd also conveniently be right next to basically infinite sources of water.
I'm happy to get into the other stuff as well, but this post is already turning into a novel!
For Mars this problem is even greater -- we already sent a lot of robots to Mars, we got tons of high quality photographs from there, better than anything we go out of the Apollo missions. The novelty has already worn off to some degree. Getting humans there will be a nice achievement, but NASA will never get a gigantic Apollo-like budget for a Mars program.
Luckily, they can contract SpaceX for a lot of it, so the cost should be more manageable. But only for getting an American flag there. Setting up a permanent Mars base would be an undertaking which is orders of magnitude larger.
Currently NASA doesn't even have any plans for a permanent Moon base! There is a recent book, "A City on Mars", which highlights the enormous difficulties of establishing a permanent or even independent Mars settlement. I haven't read it, but in this interview with the authors the prospects sound grim:
https://youtube.com/watch?v=dJqr_cCi9tM
For example, it is doubtful whether pregnancies and childhood growth work normally in low gravity environments. It hasn't even been studied properly using mice on the ISS, even though this would be perfectly possible. Maybe all the children come out severely crippled.
Then there is the constant radiation, which means everyone has to live deep underground. Then the question arises: Why not instead build underground (or underwater) cities on Earth, for a much lower cost?
They also mention things like this: Biological warfare against Earth would potentially be a lot more safe for the attackers if they sit on Mars, due to being separated by gravity wells. So the existential risk might actually increase.
They argue that we should first build a permanent Moon base before going for a permanent Mars base. There are just so many things we can learn in a much less difficult to reach environment. At least from the Moon you can fly people back to Earth in case of a medical emergency.