Why, then, do the companies only offering one class (South West, Ryan Air) tend to make a profit, while the ones with multi class (KLM, Air France, American Airlines) tend to lose money?

That video is very wrong compared to real life. For the first: it assumes a full plane. My experience is that economy tends to be full, business has a lot of available seats. For the second: it does not account for the cost of running lounges, fast track security and other complimentary services offered to business and first class passengers.

Look at the companies. Filling the plane with economy passengers and offering low service level is what brings profitability.

I believe Southwest makes its money by catering to business persons that pay more for flexible scheduling and more direct flight options rather than paying more for amenities.

iirc, southwest's seating policy allows them to turn planes around faster than airlines that assign seats.