Your assumption is that the customer will pay a infinite amount for that seat. The average customer will rarely chose the most expensive. Not saying all will not, I am saying most won't. Your Delta analogy is spot on but there is room for only one Delta. They can take less than 15% of the market for people who are willing to pay any amount for 1st class service.
I represent the other 85%. They will pay market rates. That is whatever the cheapest in the class of service they want is. So if company A is selling tickets at $150, and company B is selling them at $160 and the perceived value is the same (A-B air travel) the 85% is going to go with company A.
So now we know there is a finite amount available for profit. It's the average of the competition rates. So if everyone averages out $150 a seat then the company needs to make their profit in that $150. If the shared cost of the operation is $140 for that flight then there is $10 available to play with.
Anecdotally, every Southwest flight experience I have taken has beaten every single Delta flight I have ever flown. Both offer roughly the same result A-B air travel. I wouldn't ever pay for a first class seat, ever period dot. Some people are willing to pay for that perceived better seat most won't. Delta doesn't attract every passenger. I won't ever willingly chose Delta over Southwest if I had my choice. Not saying Delta offers bad service. But my wallet can handle Southwest, it can't handle Delta. And really the customer service at Southwest has been head over heels better at Southwest than Delta. Delta tends to attract the snobbish. The guys who complain about spending 2K for a 1st class seat and have to sit next to a deadheading pilot who got lucky enough to sit in the 2nd row with mr. big pants who thinks he makes 10x what our poor wayward deadhead makes.