I am glad he mentioned Phobos, Arthur Clarke did years ago.
A Martian elevator makes sense to me but if mankind is planning to colonising Mars then we need to know what materials are available on the Martian moons, especially if any ice remains.
If that exists it can be turned into methane and oxygen, fuel for the raptor engines, we know they have carbon and oxides which would be useful and other stuff which could be used as heat shields.
If the elevator is turned down and there is no ice on the moons, then the oxides can be turned into liquid oxygen and combined with any available light metal to make an autophage rocket using metal instead of plastic to land stuff on Mars or the oxygen can just be used as a propellant with energy beams supplying the power needed.
An evocative concept to be sure, although I've never been a big fan of generation ships - generations of trip time as opposed to build time, that is (the two do correlate). Being at heart an optimist, I figure we'll always be coming up with a better, faster way to both design and propel ships, and thus extended build projects are almost doomed to fail.
"Brains in bottles" can in theory be kept alive for quite a long time especially in a so called cold coma or topour with automatic biological repairs en route so journeys lasting several hundred years can be undertaken.