What about crazy people? What if the lottery selects a freak? Anyone who does not believe in climate change is plain nuts, and the voting system puts such people into power by the hundreds.
The goal of democracy is to represent its citizens. If its citizens are crazy in regard to climate change, then their representatives should be the same kind of crazy. That's democracy working.
A crazy person elected by lottery wouldn't represent its citizens, so that's a completely different kind of crazy.
Whether crazy or not, a lottery system is undemocratic, because it elects politicians who don't represent their constituents.
(And it would never work in practical terms, because it is too vulnerable to fraud in making that random choice. And a result would never be accepted as legitimate, because fraud would always be claimed.)