We ,sort sort of, do something like this effectively, especially in Gradient descent algorithms. A random line is simply a set of random parameters $\beta_0,\beta_1$. The gradient descent algorithm has to start somewhere looking for the optimal parameters, and the random set of parameters is one place to start.
So, in a way, we do start with a line, though we don’t draw it. Also, the algorithm itself is not exactly the one presented, of course. The instructor was probably trying to explain it without the notion of a gradient, and it’s tough. So, I’d give him a pass on a sloppy attempt.