I've missed a number of flights with AS over the years and have had no trouble being placed on the next flight. Fortunately always lucky enough that there was a seat. Usually the machine says see an agent and they hand me a seat request card... Listen for your name to clear at the gate if you can't be confirmed at check in.

Did something change where this is no longer common?

I've also had the other benefit-- door still open, MX problem, carry on only... No sign that we're leaving soon... Pull out FlightTrack and find another AS flight to same or nearby airport. This has saved me countless delay hours... Never had any trouble swapping one flight BP for another (though I believe they have a co-terminal definition when it's to a different city--ex LAX/LGB).