If we see it coming I think a very small population of humans could survive.
But it's still nice to have a "backup" planet in case of such civilization ending catastrophes.
And then there's .. you know .. because it's something to aspire to, and there's probably a bunch of interesting science we could do there. Especially with regards to life on other planets
Mars has the problem of being self-sufficient without Earth. In the near-term, if Earth dies, Mars dies because there is something like chips or biologics that they can’t make. If we have technology to survive on Mars or make them self-sufficient with only millions, then that can be used on Earth to maintain civilization on Earth for any disaster. The money spent on Mars colony pays for a lot of bunkers.
At the end of the day "self-sustaining civilization in an already inhospitable and isolated place" is probably more about showing we've already managed to find a way to eek out under pretty much any situation than it is particularly about Mars itself. To be honest life on Mars would be significantly crappier for probably decades upon decades - but at least we'd prove out the floor and our ability to raise it over time instead of assuming we'll figure things out before our assumption where we are will always be supportive to us falls through for whatever reason.
Air quality on Earth would go down a bit, but there would still be air. Food supplies would be chaotic, but some crops somewhere should survive. As for severe temperature swings on a post-apocalyptic Earth, they'd probably still be easier to handle than what NASA calculates on Mars as nightly lows of -128 C.
I realize that colonizing far away places is what Earthlings do. (Columbus, Cecil Rhodes, etc.) But the lets-go-to-Mars crowd strikes me as an extreme manifestation of Dunning-Kruger
His wealth arose from snowballing into a monopoly the purchase of Kimberley diamond leases from other European settlers over 17 years with backers money and then his own.
For that matter Columbus was in truth a failed navigator and accidental colonizer.
His intent was sail to the already colonised spice islands of the Dutch East Indies ... but he'd made a boo boo wrt the circumference of the Earth.
Columbus ended up in 1496 as Spain's colonial administrator for what's now the Dominican Republic. He did that role badly, too, but what's more important is that he did it -- creating a colonial pathway that kept widening for centuries.