Fluid dynamicist here.
If I were teaching this I would add the vorticity-streamfunction formulation before "primitive variable" incompressible Navier-Stokes in 2D. ("Primitive variables" include velocity and pressure, and not derived quantities like vorticity or the streamfunction.) It is a more theoretical approach, but it satisfies mass conservation by construction. Satisfying mass conservation in primitive variables for incompressible CFD is more complicated, and is often poorly understood. That's why I'd also have a lecture about mass conservation in the class.
Then again, Prof. Barba probably recognizes this and didn't have the time to cover this in the class.
