A solar sail of somewhat higher density but much smaller size would cast a shadow that would block only a small part of the solar disk (viewed from Earth surface) but would have to block a lot more of the solar radiation per unit area, and thus would be subject to much higher degradation. The Solar Wind is not to be trifled with. And then as you point out, you have to keep it from blowing away.
The shadow would necessarily oscillate +/-22 deg in latitude with the seasons, and probably piss off a lot of solar panel enthusiasts. Such is life.
BTW, in the 80's I designed attitude control systems for LEO and GEO spacecraft, using combinations of IR limb sensors, gyros, and electromagnetics, but those birds were orbital and had entirely different missions. Thinking about the problems of placing a quasi-stable sun-shield at L1 fascinates me.
Is there enough of earth’s magnetic field strength at GEO altitudes for magnotorquers to work for inertial wheel unloading? LEO they are common I would have assumed at GEO you would use ION thrusters or arcjets to unload the reaction wheels for attitude control. Tethers that are electromagnetic have always fascinated me. Pump current across the magfield and Lorenz force should push you to higher orbital delta V conversely pulling current using the tether as the current source sshould drag against the EM.field lowering orbital velocity down to where the atmosphere can drag a stage down. It would be a way to deprive apogee boost stages from GTO to an ocean spashdown.
These boffins want to block 1% averaged over the whole earth disk surface that would cool the planet as would a 1% loss of solar output would. From the planets point of view it’s identical loss of solar insolation. The real question is should humans mess with the earth sun energy balance at all. Carbon dioxide is a tool for Marxism, climate actually changing is a constant for the earth’s system largely driven by orbital cycles and solar output cycles in the star itself. Should the sun heat up which as it ages down the main sequence of solar fusion humans could stave off the inevitable warming for a few millennia extending our habitable period by at least that much. Eventually as the sun moves towards it’s red giant phase the earth will heat up to point oceans will evaporate and all life will end. That’s hundreds of millions of years from now. Till the oceans are in trouble.