Both Southwest, but also Ryanair are profitable. Totally possible to make money off flights.
But you have to follow the same model: use cheaper airports, a single modern aircraft type to simplify operations, high turnaround speed, charge a lot for extras.
It’s not enough to “make a profit”.
Southwest has 30B in assets and makes $441M in profit. Like most airlines it’s a miracle of modern economics and should practically be considered a charity or a nonprofit. You would make more in treasuries or corporate bonds.