If it is to preserve diverse species, then it absolutely makes sense to colonize other planets and spread the wide variety of plants, animals, and other organisms. The earth will eventually be burned up by the sun.
If interstellar space travel is ever practically attainable - and, FWIW, that is still a controversial subject - then humans absolutely will figure it out. Or rather, they will just so long as they manage to extend both their existential and economic trajectories far enough for it to matter. Ain't nobody gonna be able to develop that kind of technology from Mars; they'll be way too busy on more prosaic tasks, like repeatedly checking their domes for leaks.
So like a few million years?
Sure, but nobody today is researching the technology to survive a post-dinosaur-meteor earth. If it gets developed at all that's due to our space programs and our desire to survive on the Moon and Mars. So unless we go there we won't survive the aftermath of that theoretical meteor.
As far as why describing Mars as Earth's backup plan rubs people the wrong way, I think it's a bit like Noah's Ark. While there is absolutely the cute story about the boat full of animals, and it does make for a good story, there's also the "meanwhile literally everyone aside from a minuscule elect ends up dying" bit. Which would be nothing but sour grapes if you didn't see people holding up the possibility of colonizing other planets as a mitigating factor that implies that we really shouldn't be quite so worried about all the environmental damage that's happening here on Earth. But you do.
Put it all together, and the complete narrative starts to sound like, "Sure, we're recklessly hurtling toward turning the planet into an inhospitable shithole for billions of people, but don't worry about that, because 0.00000005% of them are going to get to live on a completely different inhospitable shithole instead!"
Which, disclaimer, I'm not wanting to pan human space exploration. But I much prefer the "exploration for the sake of adventure and discovery" narrative. If that's not good enough, the ground to retreat to should be robots, not messianic fables for rich people.