Some orgs do this, like SpeedSF in the Bay Area.
I'd recommend an HPDE after that if it's something someone would like to try (basically a track day with a coach who rides along with you every session to give advice and instruction).
That said, autox is a lot of fun and I highly recommend it to any driving enthusiast
For something run by a group like TrackMasters or GGLC or PCA, then you're definitely correct, you'll have somewhere between five and ten runs depending on how smoothly it goes, you'll work the course for a while, and you should view it as a fun day of hanging out with people who like cars and you'll get to drive a bit.
You're also definitely correct that autocross is a lot of fun and anybody who is interested in driving their car for the joy of it should try it out.
You pay $80 or whatever and get to do as many runs as you like throughout the day. Sometimes they change up the course layout halfway through.
SpeedSF's organizer for that event will go out periodically and adjust the cones and stuff. Although in my experience they get lazy halfway through the day.
But the idea is it's nothing fancy like a standard autox event. It's just a cone course to mess around on to work on handling and inputs.
This way there's absolutely no pressure. You're just there to learn your car in a small relatively controlled environment.