The earth's shadow never reaches L2 anyhow - it's only penumbra at that distance since the angular size of the earth is smaller then the angular size of the Sun
Good point. JW is at about 1.01 AU, and the solar radius is a bit over 100x the Earth radius, so there’s not really even a “shadow”.
The JW orbit semi major axis (about the L2 point) is order of 500,000 km. The radius of Earth is about 6500 km. Thus, the shadow of the Earth is extremely small compared with the excursions of JW.