First, the FAA shutdown did NOT cost the US a dime--US citizens and US airlines just didn't collect the tax, so it remained in the economy, not in the government's hands to waste.
That's only a 100% valid statement if one accepts it as a given that
any money that gets into the government's hands is wasted. I'll grant you that there is waste in government, but you'll have to grant me that it also does provide many essential services that no other entity does, or would.
I think just about everybody would agree too, that government spending and government revenue are not really related in any meaningful way. That's part of what the whole debt ceiling game was about. So, here, we have an action by a Congressman that just took $30 million/day in revenue, and moved it from the government's bottom line to the airlines' bottom line. Who benefits from that?
That the airlines could "pocket" the taxes and not pass on the difference in price to the consumer says more about the short-term elasticity of demand than airline management "spitting" at the consumer. It appears that, in the short term, tickets are pretty inelastic and, when you think of it, that makes sense--are you going to not buy a ticket, if you don't get the tax back. You were ready to buy the ticket with the tax.
Inelastic? From what I've seen, ticket prices must be made out of Silly Putty. I'd be surprised to learn that any two seats on a given flight sold for the same price. (Yeah, that was hyperbole, but you get my point.)
The elimination of the tax didn't change their costs any, and they
had been willing to sell the tickets at the lower price. So, sure, they raised their prices,
because they could. Do you think that's the sort of behavior that's designed to inspire a warm fuzzy feeling in their customers?