I patched mod-free, so you can rule out "patching with mods in" as a cause.
I'm playing modded; I can't recall how many people said they were playing unmodded on the other forum. As I recall there were some players encountering this unmodded, though - I think fullspiral plays unmodded. I had no culling or career related mods in when it happened on an older game. I'm asking on the forums if anyone with this problem is playing unmodded, but I'll also pulling out all my mods and start a new game again to test.
(edited to add: These two Simmers had the game take a Sim's job away, playing with no mods: http://forums.thesims.com/en_US/discussion/823067/my-sim-lost-his-job-in-the-patch and http://forums.thesims.com/en_US/discussion/comment/13472349/#Comment_13472349)
Yes, it happens in new games.
It happens only gradually, not right away. In my current game, it was 2 or 3 Sim days of playing another household before unemployed Barbie was made a painter. My other unemployed played Sim that I'm rotated away from is still unemployed after 5 Sim days. In 5 Sim days, I didn't see a Sim have their job removed or changed, only given when they didn't have one.
Re. "stay-at-home mom/dad": that's the same as unemployed, but is the label used for unemployed adults who have children living in the house. You won't find any of those Sims labelled "unemployed".