my initial forays into searching the controller world suggest this isn't a thing.
Unless I'm completely misunderstanding your quest, your yearning reflects a fundamental failure of Korg's marketing department.
In a bid to stay relevant as a keyboard workstation manufacturer while the whole world went inside the box, Korg took some weird sequencing stuff from this guy named Stephen Kay, raided their stillborn OASYS workstation (not that one, the one it stole the name from) and came out with the Karma, basically a much-maligned play-by-numbers workstation that attempted to bridge two worlds that hated each other - on-stage cover musicians and basement-dwelling Ableton nerds. It lasted not very long. Either you were exactly the guy that wanted standard corporate Korg workstation sounds combined with weird-ass Ableton tweakology or you hated the abomination. They were pretty much always available at steep discount from music stores.
Eventually they came out with the actual OASYS and put "Karma technology" on it because bloody hell, they were proud of it and if you were still buying keyboard workstations in 2005 you sure needed all the onboard sequencer fun'n'games you could afford. Even now, sixteen years later, the Kronos and every other big keyboard workstation you've never considered has "Karma" on it.
Thing is, though, you're still talking about mashing a button and launching a string which is far more in the realm of the Akai MPC school. See also: NI Maschine.
"Classical groove boxes" means MPC60.
You can download it for iPhone for like $10.
My buddy Dante put this song together on his phone in like half an hour.