An object at L2 absolutely could avoid the sun and stay in Earth's shadow. It would be problematical for this spacecraft, which is powered by a solar array.
> This [L2] orbit (which takes Webb about 6 months to complete once) keeps the telescope out of the shadows of both the Earth and Moon.
>What is special about this orbit is that it lets the telescope stay in line with the Earth as it moves around the Sun.
and
> Webb's position out at L2 also makes it easy for us to talk to it. Since it will always be at the same location relative to Earth-in the midnight sky about 1.5 million km away - we can have continuous communications with it as the Earth rotates through the Deep Space Network (DSN)
Muchas gracias! I've been looking for something like that. Another interesting bit I learned from the link:
> While orbits about the L2 point are inherently unstable, the orbit size is large and the orbital velocity is low (~1 km/s), so the orbit "decays" slowly. However, JWST's large sun shield, roughly the size of a tennis court, is subject to significant solar radiation pressure which results in both a force and a torque. The direction of solar force varies as the observatory's attitude changes from observation to observation. The solar torque is balanced by reaction wheels, but periodically, the accumulated momentum is dumped by firing thrusters.
If I understand correctly: JWST needs thrusters not to counteract a decaying orbit (too little energy), but to counteract accumulated momentum from solar radition (too much energy).
This simple 20 seconds video was far better than hundreds of those link. I mean I couldn't visualize the whole set up and orientation and this video made things crystal clear.