‘The future of mankind’ only rests on life beyond earth if everybody gives up on keeping earth livable. The biggest threat to humanity is humanity, and a colony on the moon or Mars isn’t going to fix that problem.
The whole idea that some sort of better type humans will travel to another planet is frankly ridiculous. We don't learn from our own mistakes - see WW1, WW2, current state of affairs. If anything, we would most likely destroy Mars and Moon much faster than how we wronged Planet Earth.
I have no idea where did you get this from :) .
Why are you thinking that's the intention?
Also the argument taken to an extreme is straightforward. We know that massive catastrophes have caused mass extinctions that wiped out most life on Earth. This is clearly visible in the fossil record. Ergo, one of these is certain to happen again. In the extremely long term, the Sun will get hot enough to kill all life on Earth in a couple hundred million years. So Humankind is doomed if we do stay on just Earth for the very long term.
There's also the side argument that "right now" appears very close to making it possible. If Starship and the attempt at reusable rockets were to disappear there's nothing guaranteeing the progress of space technology. It's easy to see a future where we do some more footprints on the Moon and maybe later some footprints on Mars, but no permanent off-earth colony is ever established. Eventually we give up on creating such bases because they're deemed too expensive and "robots do it better" as our robotic technology gets better and better. The technology of how to do reusable vehicles is eventually forgotten over several generations and then we never leave Earth again. (Eventually resulting in our destruction.) This is a "why not now?" argument.
The path to making colonies on Mars or the moon self sustaining is not even clear at this point. Starting with the simplest of fundamentals - air, water, food and protection from cosmic radiation. The prospect of growing a technical civilisation on either body, that can develop further without support from Earth is remote.
By contrast, to make earth unlivable basically means nuclear war. Climate change won't do it, biological pandemics won't do it, nor will fossil fuel exhaustion or any other localised event.
On the worst day possible on earth (global radiation aside) its still a million times better to live on than anywhere else.
In terms of global radiation making the world uninhabitable even then tiny, non-sustaining pockets of humanity would survive. At least until their life-support systems failed.
Lastly, I wonder at the need for "the future of mankind" as a goal at all.
That said, ‘colonizing’ the artic or shallow ocean floor probably accomplishes most of the benefit for a subset of the cost and risk.
The only real benefit I see to space over the attic is that off worlding mining and production could be a benefit in itself.
If all the water on mars were melted, it would cover the planet in 100ft of water [1]. With the safe assumption of any kind of water recycling, and indoor habitats, there's plenty of water on mars, which also means there's plenty of oxygen on mars (along with the 95% CO2). The atmosphere is 3% nitrogen, and 1.6% argon, which means there's plenty of air on mars. You don't have to fill the sky with air, just the buildings.
[1] https://marsed.asu.edu/mep/water#:~:text=Taking%20what%20can....)
(We should probably start with the moon, though)
An off planet colony would likely face the same issues of sustainability as soon as different factions start competing for resources.