I suggest that not only does ID not make us safer, it makes us less safe.
As an example, a teacher of mine had escaped Cuba in a small boat in the 1960s. This made him a criminal fugitive in the eyes of the Cuban government. A few years later, he was on a Miami to Tampa flight that was hijacked to Havana. (in Flyertalk parlance, MIA to TPA; irrops; HAV) He hid/destroyed his documents when he realized that the flight was crossing the Florida Straights. He avoided recapture partly because it was plausible for an American to be traveling without ID.
To leave the realm of anecdote and cite something verifiable, Petty Officer Robert Stetham was killed by the TWA 847 hijackers because they found his U.S. military ID. Leon Klinghoffer was singled out by the Achille Lauro highjackers because his U.S. passport listed his birthplace as Israel.
Do you think that ID made those folks safer? Once the scanning check point has confirmed that I don’t have a weapon or an explosive, why does it matter whether I’m John Smith or Joe Bloggs?