First off, in the 'backup' argument Mars isn't meant as a bomb shelter for the human species. It's a shelter for human civilization, which is a higher level and much more fragile thing. Think about the libraries of the Arab world, not "The Arabs have to exist or humanity could die out when Rome falls!"
The other argument is as a launchpad. Mars is halfway to anywhere in the same way low earth orbit is, but Mars comes with exploitable resources. In the future, we can debate whether those resources are better exploited by enormous robotic fleets and autonomous systems or with humans on site, but with current tech readiness levels nothing beats in-situ humans for adaptability. A strong Martian industrial base will produce huge returns for terrestrial society. From dense high-throughput communication networks to massive orbital manufacturing depots to idyllic ring-world habitats, everything we want to put in orbit is cheaper starting from Mars.
They're only just proving this out, but one of the theses behind Made in Space [0] is that there are all sorts of microscopic manufacturing processes - optical components and chips being the most obvious - which would benefit from manufacturing in an ambient gravity between 0-1g. Which you simply can't get on Earth.