I suspect the TSA would be charged enforcement of FAA rules regarding these things, don't you?
Nope. Story time.
Years ago, I was flying for a small freight feeder. I was based in Minot, ND, and flew back and forth between there and Sioux Falls, SD. The TSA was on the ramp doing random inspections. They climbed on the airplane, went into the cockpit and found our crash axe, a required piece of equipment per the TCSD (type certificate data sheet). We flew these airplanes around single pilot. They were converted passenger turbo props. They pulled it out of it's pouch, brought it to the office, and told us we couldn't have weapons on the aircraft, because it would make it easier to hijack the airplane. Mind you, we flew around with just one person on the airplane, who was going to hijack the damn thing? The TSA tried to ground us, the entire airline. The Chief pilot and director of operations tried explaining to them that it was a required piece of equipment to cut our way out if the worst should happen, and they weren't budging. It ended up with a call to Washington, to get a higher up with the FAA and the TSA to have a sit down, and say "look, we regulate the airplane, you handle the security of the passengers, capiche!" It was a huge political pissing match that put a stop to an entire airline for about 8 hours. It was a damn good thing it happened in the middle of the day when most of our airplanes were parked anyways.
The TSA is an over reaching, under achieving government agency that is nothing more than "it makes my feels better." They suck at what they do, horribly. I regularly get my hands swabbed for explosives. I've been through a 10 year FBI background check just to sit up front, the same type for a secret clearance yet THE TSA feels it's necessary to make sure I'm not building IED's. I also shoot as a hobby. One of our overnights is right across the street from an indoor range. That's always a pleasure when I rent a pistol, and go through security the next day.
To conclude, the TSA is tasked with security, nothing else. The FAA is in charge of everything else, and some security as well, just not passenger screening.