Lets completely remove liability from the picture. The airlines are simply not responsible for anything that happens to the passengers, and as a result they don't care. Seatbelts, oxygen systems, escape slides - who needs them? Flight recorder? In the trash. FAA staff are cut to 20% of current levels and are mainly concerned with keeping things running smoothly (their pay is docked across the board when flights are late) and the NTSB is defunded.
Daily operating costs go down a bit. We've removed a couple of hundred kilograms of safety equipment and that will make a small dent in the fuel bill. But only a small one. The mini-Starbucks installed where the over-wing exits used to be weighs about the same. Costs related to FAA-mandated paperwork are eliminated, but only temporarily.
There's less training for the cabin crew, but we can replace the evacuation training with sales training to bump up profits from regular and duty-free sales. Hooters re-enters the air carrier industry and does quite well, as cabin crew can be transferred directly from the restaurants skipping all that troublesome FAA qualification business.
But we find that maintenance costs don't go down. Costs plummet for the first year, but then go back up as management realizes that an airplane on the ground because it won't start costs a lot more in losses than maintaining it properly to begin with. And the one that crashed on takeoff was a complete loss on the airframe ($100mil and up) along with a dent in sales for the following 6 months.
Flight recorders are reinstalled as they are actually rather useful to predict if something is going to break. It can then be replaced by the night shift. The FAA paperwork we stopped doing is also quite useful for deciding wether or not to overhaul the wheel bearings with winter-weight grease, as we wrote down when it was last done.
The flying public will see maybe 10% reduction in fares. Remember that a very large percentage of ticket costs are taxes. For example, flying out of the UK costs up to £194 just for taxes. Wether the public will accept a large decrease in oversight for a small decrease in fares, or wether the long-term accident rate actually goes up or down will be a very interesting social experiment.