Used to work for a company who had to fly in right after hurricanes to set up shop. Any Corporate flight department for a major insurer is pretty well experienced in these sorts of operations. State Farm used to have routers and telephone systems to handle 5,000+ lines to a sat link which were palletized down to fit in a fleet of Learjets. That was back in the 1990's, they probably handle 25,000+ lines today and have their own sat link.
Hurricanes are flat. Over ~ 25,000 feet they are nothing other than a curiosity as you fly over in smooth air. On arrival the most significant hazard is debris on the runway. The winds (the day after) are steady in one direction and if you don't like the landing weather, wait 6 hours and the winds will have shifted.
As others have mentioned the infrastructure is too destroyed and folks are too wrapped up in other projects to expect a 121 flight to be operated in or out the day before, or day after.
Again, the airplane is happy. The problem is the infrastructure on the ground & debris. If you tear up your airplane by sucking, or running over, debris you are stuck. Forget getting parts & mechanics. Folks do not come to work when their car is floating in the pool. Worse, after Katrina the locals showed up at the airport and started taking every part that they thought was worth any money. One client of mine was very worried as they saw two of their Corporate jets get picked clean using their satellite imagery.
Flameouts? We ran with the ignition on through some of the very wet bands which radiate out from the storms, but that is not much different than your procedures in any heavy rain. You will be surprised how steady the wind is.
Nobody is going to dispatch you in the thick of the storm. The day after is no big deal. Three days after and your biggest problem is finding a cold beer and dealing with the mosquito's that take advantage of the warm weather and lots of water.
By the way, if anyone sees a Airboat in Louisiana powered by an Garret/Allied Signal TFE-731, PM me and we can probably get you a reward. There are a bunch of airplane motors on Airboats, ever wonder how they got there?