This is the way I taught my math classes at two different universities while I was in grad school. I even used chalk. As I worked through the proof or problem, writing out everything in full meant we could have an interactive flow where if I lost them they could complain. And they did! I would get comments/questions even if I was writing on the board, with my back turned. I don't recall time management ever being an issue, and my practice was to just work through everything beforehand.

It was one of the most satisfying/gratifying experiences of my entire life.

(Dirty secret: I only deeply learned vector calculus/linear algebra when I taught it.)

(Edit: grammar stupidities)