There are elements of both at work. Companies hire people because they need to, but they expand services because they can afford to (and presumably because it gives them an edge over the competition).
And your point on "all else stays equal" omits two important variables:
-The first is that, at present, there are a lot of markets where one or more LCCs are heavily restricting fares. A case in point would be to compare a flight to/from Des Moines to one to/from a similar-sized market with LCC involvement. Yes, this is going to be more extreme than your average market (DSM is one of the worst in the country), but it's an example. In a lot of cases, this has led to unnaturally low fares as the various operations try to run one another into the ground fighting for market share. I think Cornelius Vanderbilt had more than a few frustrated comments about having to do this with the Pennsy.
-The second is that, absent at the very least ULCCs, fares rising won't necessarily run off enough business to offset the higher fares. This wouldn't be an issue, except that more often than not, carriers have tended to use the added revenue to either run extra frequencies throughout the system/expand operations to new markets (which can beget a virtuous cycle of market growth) when they weren't trying to reduce debt.
You're omitting one important point: Airlines aren't expanding capacity today, in fact they are reducing capacity, specifically because they want to keep fares high.
When fares are high, people don't fly as much. Think of a typical demand curve. When price is high, quantity demanded is low. When price is low, quantity demanded is high.
What has hurt airlines in the past decade has mainly been rising fuel costs. The result of this is that we are starting to see a reduction in air service because airlines can't raise fares high enough to cover those costs, because doing so does, in fact, drive business away. The markets able to sustain the higher air fares are seeing a reduction in capacity as airlines find it more profitable to fly a smaller plane full at higher fares rather than a large plane mostly empty, or even a larger plane full with low fares. That's why you see 50- and 70-seat regional jets on routes that, 20-30 years ago, saw DC-10s.
The airlines aren't using those higher fares to expand service. They're using them to stay alive. But smaller planes require less staff than larger ones, so that's a net reduction in staff (and, regional jets are also flown by airlines that pay their staff less, but that's outside the scope of this discussion).
The existence of low cost airlines in this country has led to a significant expansion in the number of people flying, because they can now afford to whereas before, they couldn't. Airlines like Southwest and JetBlue (and Spirit) have enabled more people to fly. True, they have taken passengers from other airlines, but there are still plenty of people who won't fly low-cost carriers. If you got rid of Spirit, some of those passengers would head over to other airlines (resulting in a small capacity bump on those carriers), but others would simply not travel at all. With a reduction in passenger traffic, there would necessarily be a capacity reduction. That also means a reduction in employment.
As for your initial example of DSM vs. similar markets with LCCs, what is the total number of flights to DSM vs. that hypothetical "similar" market? In many cases, LCCs entering a market has led to an increase in traffic. Also, your comment about airlines trying to "run one another into the ground fighting for market share" may have applied 10-20 years ago, but is no longer the case. This goes back to my point earlier. Airlines are being a lot smarter with their capacity, realizing that they can't afford to get into a fare war like they used to. That's one of the things that's driving the latest push for industry consolidation. The airlines themselves realize there are too many of them out there, and there's too much capacity, and that's depressing fares and profitability.
Bottom line, I don't see any real argument that says that an airline like Spirit shutting down will lead to a net increase in employment in the airline industry.