Note that the drawings don't always have to be tight or even too on model. The important thing a storyboard artist should concentrate on is creating and telling the story. Continuity, staging, pacing and entertainment. There are other departments to refine visually what the story artists write.
Most old cartoons were timed to musical tempos. Cartoons were meant to appeal to the senses. They aim to look good, sound good and move pleasantly.
Disney had a luxurious production system. His storymen would draw rough boards first just to get the ideas and rough continuity down, then they would draw tighter boards with rendering and even color - sometimes just to impress Walt and help sell their ideas.
Then they would time them and shoot them on test reels to see if the continuity was working. They left it open at every stage to lots of changes and revisions according to how well the stories worked visually and rhythmically.
They sculpted their stories with groups of people, constantly tweaking and changing and revising. Of course all this was very expensive at Disney's since it was always open to changes. The other studios streamlined the process.
Even the folks who rebelled against Disney stylistically still used the basic logical cartoon production system.
For lots more details on great story art from your favorite cartoons, click the link at the top and go through many pages of classic storyboards.