Yup, Earth’s Moon is far more amendable to a beanstalk for exactly the reasons you mention. Mars is an edge case, but you’d still have most of the same stability issues. The Martian atmosphere is a wee bit thin for aircraft to be much help, but it also resists a catapult less and a rotovator could reach lower without experience significant drag. I suspect the Moon is uniquely qualified for a beanstalk. However, there’s no need for an L2 beanstalk. A transfer orbit from L1 to L2 takes a pittance of energy. If you’re thinking of the L2 beanstalk for launching elsewhere in the solar system, that’s where a smaller momentum exchange tether is more useful as it can be somewhat aimed on an efficient orbit.
Most likely any use of Lunar water would remain on the Moon for local projects. The rest of the Solar System has plenty of water-ice.
You’d be better off placing automated solar stations at the Earth-Moon L2 point. Alternatively you could place them at the L4 or L5 points for easy beaming to LEO satellites.
