Imagine we saw the mother of all asteroids incoming. It was large enough that it would almost certainly extinguish all life on Earth, and even throwing every nuke we had at it would barely scratch it. Yet we also saw this from relatively far away - let's say about 10 years. It seems exceptionally probable that by the time it got here we'd have established colonies on both the Moon and Mars.
The main issues are not technological in nature anymore - but economic. We put a man on the Moon more than half a century ago. The fundamental requirements to convert a short-term stay to a perpetual one are really not that great: food, home, maintenance. Water, which both the Moon and Mars have - in abundance, can be used to generate oxygen and fuel - taking care of near to all essentials. So what you really need is the infrastructure - housing, agriculture/ranching, industry, and recreation.
And that's all entirely viable, but has a very high price tag. SpaceX is trying to tackle this problem by bringing the price of space launches as close to $0 as possible, but even once that is achieved - the actual materials necessary to achieve this, even at cost, will be very substantial. And without some imminent danger, there will be people claiming that we shouldn't colonize another planet because everything here isn't perfect, which it never will be.
The cost of establishing a Mars colony is so high that the money is probably better spent in a different way. An asteroid defense program (using big Starship rockets), for example.
I don't see it as just a thing of ensuring the species, a la Elon. But even if you do, there are countless other scenarios it guards against - self annihilation via nuclear weapon, bioweapons or bioweapons research accident, any of the zillion astronomical phenomena that could screw out atmosphere (like a gamma ray burst, as hypothesized for a cause of one the great mass extinction events), a super volcano explosion (again hypothesized as the cause for a previous mass extinction event), and so on.
I think us doing something on a huge scale that people could actually be proud of, even if they didn't directly take part in it, is something that could be good for society on a much more broad level. We've achieved a lot in "our time", like the internet, but oddly enough it never really felt like an achievement - by contrast the Moon landing is something who got to experience even today speak fondly of.
And in any case, the exceptional cost is only for a truly self sustaining and independent planet, as would be needed in case of a catastrophe. We can "safely" start with a dependent system where various industrial goods can be shipped to e.g. Mars, and ideally even back to Earth as Mars' industry starts to come to life - even silly things like trinkets from Mars' surface, alcohol brewed on Mars, etc. will all be simple and valuable trade goods.
Getting things started will be done for a relatively low cost. Fortunately, thanks to interests like SpaceX this initial seeding need not even come from the public at large. And hopefully once something is there, that will inspire more people to see what could be there.
Getting the first human group on Mars does indeed sound like an inspiring goal. But getting the second or tenth group there sounds quite uninspiring. Remember, after some people landed on the Moon, the novelty wore off rapidly and taxpayers rapidly lost interest.
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:
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.
This isn't quite what happened with the Moon landing at all. Even throughout the Apollo program, it was relatively unpopular. A large minority were exceptionally happy about it, but you have to remember this was also the era of Vietnam, race politics, and more. And that led to a lot of opposition to things like spending billions of dollars on space that could be spent on [x]. Like today. But what happened is that after Nixon defacto cancelled the space program in 1972, support for space began to grow dramatically. It turns out that people looking back didn't quite realize what they had, until it was gone.
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!
The main issues are not technological in nature anymore - but economic. We put a man on the Moon more than half a century ago. The fundamental requirements to convert a short-term stay to a perpetual one are really not that great: food, home, maintenance. Water, which both the Moon and Mars have - in abundance, can be used to generate oxygen and fuel - taking care of near to all essentials. So what you really need is the infrastructure - housing, agriculture/ranching, industry, and recreation.
And that's all entirely viable, but has a very high price tag. SpaceX is trying to tackle this problem by bringing the price of space launches as close to $0 as possible, but even once that is achieved - the actual materials necessary to achieve this, even at cost, will be very substantial. And without some imminent danger, there will be people claiming that we shouldn't colonize another planet because everything here isn't perfect, which it never will be.