Result: Each air change costs around .62 of a gallon of gas (assuming perfect efficiency). Or around $2.50 - let's call it $10 to account for inefficiency. In contrast it costs around $10,000 per hour (not including salary) to fly a large plane.
Saying they don't circulate air to save money is ridiculous.
You can work this out my looking at an an airlines profits, and how many flights they did a year.
By my calculations a smaller 70 passenger plane would use around 1/5 the energy per air change, and they typically fly for only an hour or two. So for an air change every half an hour it would cost them around $4 per flight (and that's assuming low efficiency). So $100 profit is still possible (although I have to say that that's amazing low, but I'll take your word for it).
In this case, airlines are a business designed not to have runaway profit. The unions are very powerful. They know how much money the airline is making a demand large percentages of it. They leave a nominal profit for key investors but shareholders as a whole are regularly left out. In fact, if you look at the entire history of commercial flight, the industry is break-even.
That's not a fluke. If costs were higher, unions would get a smaller cut. If costs were lower, unions would take more.
Philip Greenspun has some great writing on this, as well as the old, shortlived but fantastic Enplaned blog.