There are a ton of small Yahtzee clones like Zombie Dice, but I recommend Martian Dice as the best of the bunch. These fall into the Spot It/Jungle Speed category of games you'd play when waiting in line or killing time, not as an actual warmup.
I played it once (plus expansion), and the rerolling seemed more prominent than it otherwise would be in a typical game. You're right in that it is much more like Kingsburg.
I think I'm tired of / done with the games that use dice in this manner, however, moreso than I am with Yahtzee games. But, tired nonetheless of both. King of Tokyo gets a pass in that it's elegant and SHORT.
I'm curious to see if the expansion causes more re-rolling. I bought all the extra stuff at PAX East when Game Salute was clearing out on Sunday afternoon, but haven't opened that stuff yet. I'm only familiar with the base game. As for tiring on those types of games, I'd sort of agree with you, but I love Castles of Burgundy too damn much. I suppose that's more dice driving actions rather than dice driving worker placement though.
I got a relatively nice Yahtzee set with a dice cup and a felt dice tray when I was still in middle school. Still have the dice kicking around, mixed in with my RPG dice.
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My family wasn't into board games. I mainly played Sorry, Monopoly, Life, and all that junk.
But if you were born in the 80s, I assume you played Pop-O-Matic Trouble with your family.