This is not a launch alternative. This is maybe a get-to-geostationary alternative. This is really an L1-base station-keeping alternative. The Moon-side cable anchors the base in the radial direction, and the Earth-side cable keeps the Moon-side cable in tension, and lowers the fuel cost of reaching orbits that the cable crosses.

The math suggests that putting a rendezvous base 41% of the way from the Earth to the Moon would be better than extending more cable down into Earth's gravity well. Meeting the cable closer to Earth would always cost more fuel. You could save fuel to GEO by meeting the cable further out, and then climbing back towards Earth. But every bit of extra cable you add on the Earth side increases the tension at the max-tension point at L1.

So just stop at the lowest-fuel rendezvous height, and add enough mass there to keep the whole cable in tension, but not enough to break the cable at L1.

(The L1 point is at about 84% the distance from Earth to Moon.)

The same type of material would likely be usable to anchor a halo relay satellite slightly beyond Earth-Moon L2, just by bolting it down to the darkest antipodal point on the dark side of the Moon.