>”Apparently it IS possible to fly without an ID (for some people, at least).”
Yup. It is. My wife and I parked at my work, walked to the local airport, hopped on the puddle-jumper twin-prop and got to Anchorage for a flight to Hawaii. Where my wife realized she didn’t have her wallet with her.
GET TO TSA EARLY. It will take a while. Bone up on every place you’ve ever lived (address, etc) and every car you own (make, model, year, etc). Once they start interviewing you, you can’t refer to your phone, talk to your traveling companion, and they ignore the companion who’s trying to point out that you look exactly like the doctor you’re claiming to be, on the hospital’s website. They’re only interested in you confirming info they can access through certain governmental sources (but not get creative like the State’s Professional License search function, or what her DEA number is), although her vaccine card did help a bit.
You can go the DMV and claim you lost your driver’s license and get a new one. Now you’ve got an extra one to hide in your usual carry-on bag.
Post-9/11, I went to the DMV and got a state ID for $10. Looks like a driver’s license and it works the same for getting through TSA (although not with the rental car agency).