Economy class seats generally speaking will earn the least amount of money to an airline, so this idea would actually likely be the most unprofitable option.

To make it truly profitable, you convert the plane to First class, but then you'll need to fill all those seats.

Wendover Productions had a great video on this a few years ago: https://www.youtube.com/watch?v=BzB5xtGGsTc

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.

You still have to fill it. It would probably be easier to fly 3 or 4 A321LR instead depending on demand.