To answer your question at the end, I think it comes down to risk vs reward.
> why not just wait
We can wait, that's an option that's less costly, but more risky.
If something planet-scale happens while we wait on a single planet, the entire species would not make it. If we don't wait and pay the extra price of going to multiple planets sooner, the species has a better chance of survival in the case of such an event.
On the other hand, the list of disasters that would leave Earth less inhabitable than Mars without also destroying any Mars colony is pretty rare. If human survival is the primary goal, it might be best to work on perfecting that sort of self contained habitat somewhere in the remote parts of Earth.
I suspect even Musk doesn't believe this. I've never read/heard him talk about our species being wiped out. He usually prefers the much more vague concept of end of our "technological civilization".
I think there is a strong humanitarian argument for making life multi-planetary in order to safeguard the existence of humanity in the event that something catastrophic were to happen, in which case being poor or having a disease would be irrelevant, because humanity would be extinct.
I think there are really two fundamental paths. History is going to bifurcate along two directions. One path is we stay on Earth forever, and then there will be some eventual extinction event. I do not have an immediate doomsday prophecy, but eventually, history suggests, there will be some doomsday event. The alternative is to become a space-bearing civilization and a multi-planetary species, which I hope you would agree is the right way to go.
Either we spread Earth to other planets, or we risk going extinct. An extinction event is inevitable and we’re increasingly doing ourselves in.
... along the timescale of millions of years. Dinosaurs lived for over 100 million years. What a load of rubbish.
That's a reasonable view too, IMO. If our technological civilization suddenly collapses for any reason, most of us here, and double-digit percentage of world population, will die a painful death. The lives of whoever remains won't be pleasant, and the technological civilization probably won't arise again for tens of thousands of years (given how we've used up all easily accessible energy sources).
Life is very resilient. Humanity as we know it is not.