I think I just figured out why people create PDF forms that don't let you save the contents typed in, as annoying and unhelpful as that is.
I'll bet they're trying to protect us from ourselves, reasoning that we will fill out the form with sensitive information such as SSN, date of birth, etc., and we can't be trusted to store the saved file securely.
So they don't let us save it. It feels a little nanny-state, but I guess it's at least well-intentioned.
Edit: I think there could be a niche market opportunity here for a little piece of software that allows writing anywhere on a PDF and ignores the do-not-save flag.