I doubt the billionaires themselves would personally go move to mars, or if they did, they'd only be doing so out of foolishness without realizing what they were signing up for, and I say this as someone who views space colonization as a vitally important goal for the future. The fact is that mars is not a habitable planet like earth is, it's missing a number of different parameters, and terraforming it isn't something we have anywhere near the economic capacity or technology for, and would still take centuries to millenia to complete if we started.

That doesn't mean we can't colonize it, we don't quite have all the technology there but I would bet we aren't that far from developing it, and we almost certainly could make a base to at least expand into a proper colony as we developed the needed tech for self sufficiency, it just would be prohibitively expensive with our current lack of space infrastructure and manufacturing. But it does mean that any colony we build there is going to be a small bubble of artificial habitability in an inheritly deadly environment, and is going to feel that way until the colony gets very developed and expansive enough to fit things like gardens and other amenities, which would probably be a long way off, especially considering mars would be our very first space colony or at least one of the first if we decide to go for the moon first or something.

I can imagine the best analogy on earth to a very early space colony made with near future technology would be living in a submarine, in cramped, mostly artificially lit conditions with a small number of other people , surrounded by complex and expensive machinery that needs to be constantly maintained as it's failure could rapidly lead to death. With the difference that if something goes wrong, help would take years to many months at the very best to reach you, there is no returning to "shore" without potentially years of waiting and planning, and even the gravity is different. Not the sort of life I can imagine anyone not very dedicated to the idea for one reason or another signing up for, least of all a billionaire. Living on earth would be easier even if you were living in a bunker after firing every nuclear weapon and burning every scrap of coal in the ground; at least then you'd be able to extract oxygen and water and potentially usable soil from the outside environment, and have comfortable gravity.