Not really. Case in point - the Moon. It's absolutely massive, like several orders of magnitude larger than any satellite we've ever launched, and when it happens to line up just right between the Earth and the Sun, the umbra is only like 150km wide.

If the sun was a point light source that is accurate. It is not though. You would need an absolutely enormous sat to cast a shadow that would actually affect our weather. Plus you would also need a way to keep it in position as the solar radiation would push it out of position quickly.

That's... Not how shadows work... Or orbital mechanics... Or economics...

His suggestion, impossible as it is at the current stage of human development, is slightly less impossible than this.