At least up to the 1970s, very cheap standby air fares used to be fairly common. Often they were restricted to under-25s and called "youth fares", and if I remember correctly were exactly as described in the OP -- you showed up at the airport and got put on a waiting list, and if there was an open seat at the last minute, you got it cheap. I haven't heard of such fares for a long time. I don't know why they stopped being offered.