I don’t think this is that difficult. Don’t bring a gun to an airport. If you must for transport, arrange for a professional to do it for you. If you can’t afford that, you don’t have a good reason to travel with a firearm.
I somewhat disagree with it being straightforward. 99% of the time it can be, but there's some notable edge cases that are missing from that website. There's been a handful of cases where people have flights with connections through NY/NJ, miss their flight (or have to divert to NY/NJ for mechanical problems and transfer to a new aircraft), collect their luggage that has a firearm in a locked container, and then get arrested for possessing a firearm without the mandatory NY/NJ state permit (which IIRC are only available to residents)
Anyone carrying a gun in an airplane, or any public transportation method is either a security officer, or a criminal. There are no good reasons as a regular-ass Joe to carry your Glock in an airplace. Not a single one.